Merge pull request #1 from AndreasFMueller/master

update

56 files changed, 3599 insertions, 0 deletions

diff --git a/vorlesungen/slides/5/Aiteration.tex b/vorlesungen/slides/5/Aiteration.tex
new file mode 100644
index 0000000..3078c55
--- /dev/null
+++ b/vorlesungen/slides/5/Aiteration.tex
@@ -0,0 +1,59 @@
+%
+% Aiteration.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Iteration von $A$}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.34\textwidth}
+\begin{block}{$\varrho(A) > 1\uncover<4->{\Rightarrow \|A^k\|\to\infty}$}
+\uncover<2->{%
+Eigenvektor $v$, $\|v\|=1$, zum Eigenwert $\lambda$ mit $|\lambda| > 1$}
+\uncover<3->{%
+\[
+\|A^kv\| = |\lambda|^k\to \infty
+\]}
+\uncover<4->{$\Rightarrow \|A\|^k\to\infty$}
+
+\end{block}
+\end{column}
+\begin{column}{0.63\textwidth}
+\begin{block}{$\varrho(A) < 1\uncover<12->{\Rightarrow \|A\|^k\to 0}$}
+\uncover<5->{%
+$A$ setzt sich zusammen aus Jordanblöcken:
+\[
+J(\lambda)^k
+=
+\renewcommand{\arraystretch}{1.2}
+\begin{pmatrix}
+\lambda^k&\binom{k}{1}\lambda^{k-1}&\binom{k}{2}\lambda^{k-2}
+	&\dots&\binom{k}{n-1}\lambda^{k-n+1}\\
+    0    &\lambda^k&\binom{k}{1}\lambda^{k-1}
+	&\dots&\binom{k}{n-2}\lambda^{k-n+2}\\
+    0    &    0    &\lambda^k&\dots  &\binom{k}{n-3}\lambda^{k-n+3}\\
+ \vdots  & \vdots  & \vdots  &\ddots &\vdots\\
+    0    &    0    &    0    &\dots  &\lambda^k
+\end{pmatrix}
+\]}
+\uncover<6->{Alle Matrixelemente konvergieren gegen $0$:}
+\[
+\uncover<7->{\binom{k}{s} \le k^s}
+\uncover<8->{\Rightarrow
+\underbrace{\binom{k}{s}}_{\text{\uncover<9->{polynomiell $\to \infty$}}}
+\underbrace{\lambda^{k-s}}_{\text{\uncover<10->{exponentiell $\to 0$}}}
+}
+\uncover<11->{\to 0}
+\]
+\end{block}
+\end{column}
+\end{columns}
+\uncover<13->{%
+{\usebeamercolor[fg]{title}Folgerung:}
+Es gibt $m,M$ derart, dass
+$m\varrho(A)^k \le \|A^k\|  \le M \varrho(A)^k$
+}
+\end{frame}
diff --git a/vorlesungen/slides/5/Makefile.inc b/vorlesungen/slides/5/Makefile.inc
new file mode 100644
index 0000000..4ca3de4
--- /dev/null
+++ b/vorlesungen/slides/5/Makefile.inc
@@ -0,0 +1,44 @@
+
+#
+# Makefile.inc -- additional depencencies
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+chapter5 =								\
+	../slides/5/verzerrung.tex					\
+	../slides/5/motivation.tex					\
+	../slides/5/charpoly.tex					\
+	../slides/5/kernbildintro.tex					\
+	../slides/5/kernbilder.tex					\
+	../slides/5/kernbild.tex					\
+	../slides/5/ketten.tex						\
+	../slides/5/dimension.tex					\
+	../slides/5/folgerungen.tex					\
+	../slides/5/injektiv.tex					\
+	../slides/5/nilpotent.tex					\
+	../slides/5/eigenraeume.tex					\
+	../slides/5/zerlegung.tex					\
+	../slides/5/normalnilp.tex					\
+	../slides/5/bloecke.tex						\
+	../slides/5/jordanblock.tex					\
+	../slides/5/jordan.tex						\
+	../slides/5/reellenormalform.tex				\
+	../slides/5/cayleyhamilton.tex					\
+									\
+	../slides/5/spektrum.tex					\
+	../slides/5/normal.tex						\
+	../slides/5/unitaer.tex						\
+									\
+	../slides/5/konvergenzradius.tex				\
+	../slides/5/krbeispiele.tex					\
+	../slides/5/spektralgelfand.tex					\
+	../slides/5/Aiteration.tex					\
+	../slides/5/satzvongelfand.tex					\
+									\
+	../slides/5/stoneweierstrass.tex				\
+	../slides/5/potenzreihenmethode.tex				\
+	../slides/5/logarithmusreihe.tex				\
+	../slides/5/exponentialfunktion.tex				\
+	../slides/5/hyperbolisch.tex					\
+	../slides/5/chapter.tex
+
diff --git a/vorlesungen/slides/5/beispiele/Makefile b/vorlesungen/slides/5/beispiele/Makefile
new file mode 100644
index 0000000..05bd5b5
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/Makefile
@@ -0,0 +1,32 @@
+#
+# Makefile
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	kern bild kb kombiniert.jpg leer.jpg drei.jpg
+
+kern:	kern1.jpg kern2.jpg
+bild:	bild1.jpg bild2.jpg
+kb:	kernbild1.jpg kernbild2.jpg
+
+JK1.inc:	kernbild.m
+	octave kernbild.m
+
+kernbild1.png:	JK1.inc common.inc kernbild1.pov
+kernbild2.png:	JK1.inc common.inc kernbild2.pov
+bild1.png:	JK1.inc common.inc bild1.pov
+bild2.png:	JK1.inc common.inc bild2.pov
+kern1.png:	JK1.inc common.inc kern1.pov
+kern2.png:	JK1.inc common.inc kern2.pov
+kombiniert.png:	JK1.inc common.inc kombiniert.pov
+leer.png:	JK1.inc common.inc leer.pov
+drei.png:	JK1.inc common.inc drei.pov
+
+%.png:	%.pov 
+	povray +A0.1 -W1920 -H1080 -O$@ $<
+
+%.jpg:	%.png
+	convert -extract 1080x1080+420+0 $< $@ 
+
+clean:
+	rm -f *.png *.jpg
diff --git a/vorlesungen/slides/5/beispiele/bild1.jpg b/vorlesungen/slides/5/beispiele/bild1.jpg
new file mode 100644
index 0000000..879fae8
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild1.jpg
diff --git a/vorlesungen/slides/5/beispiele/bild1.pov b/vorlesungen/slides/5/beispiele/bild1.pov
new file mode 100644
index 0000000..fd814f1
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild1.pov
@@ -0,0 +1,13 @@
+//
+// bild1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/bild2.jpg b/vorlesungen/slides/5/beispiele/bild2.jpg
new file mode 100644
index 0000000..2597c95
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild2.jpg
diff --git a/vorlesungen/slides/5/beispiele/bild2.pov b/vorlesungen/slides/5/beispiele/bild2.pov
new file mode 100644
index 0000000..6e3c6dd
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/bild2.pov
@@ -0,0 +1,17 @@
+//
+// bild2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j11, 0.7 * at, orange1)
+arrow(O, j12, 0.7 * at, orange1)
+ebene(j11, j12, orange1)
+
+arrow(O, j21, at, orange2)
+gerade(j21, orange2)
+
+
diff --git a/vorlesungen/slides/5/beispiele/common.inc b/vorlesungen/slides/5/beispiele/common.inc
new file mode 100644
index 0000000..ffcff60
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/common.inc
@@ -0,0 +1,134 @@
+//
+// common.inc
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+#version 3.7;
+#include "colors.inc"
+
+global_settings {
+	assumed_gamma 1
+}
+
+#declare imagescale = 0.25;
+#declare O = <0, 0, 0>;
+#declare at = 0.02;
+
+camera {
+	location <3, 2, -10>
+	look_at <0, 0, 0>
+	right 16/9 * x * imagescale
+	up y * imagescale
+}
+
+//light_source {
+//	<-14, 20, -50> color White
+//	area_light <1,0,0> <0,0,1>, 10, 10
+//	adaptive 1
+//	jitter
+//}
+
+light_source {
+	<41, 20, -50> color White
+	area_light <1,0,0> <0,0,1>, 10, 10
+	adaptive 1
+	jitter
+}
+
+sky_sphere {
+	pigment {
+		color rgb<1,1,1>
+	}
+}
+
+#macro arrow(from, to, arrowthickness, c)
+#declare arrowdirection = vnormalize(to - from);
+#declare arrowlength = vlength(to - from);
+union { 
+	sphere {
+		from, 1.0 * arrowthickness
+	}
+	cylinder {
+		from,
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		arrowthickness
+	}
+	cone {  
+		from + (arrowlength - 5 * arrowthickness) * arrowdirection,
+		2 * arrowthickness,
+		to,
+		0
+	}
+	pigment {
+		color c
+	}
+	finish {
+		specular 0.9
+		metallic
+	}
+}
+#end
+#declare r = 1.1;
+
+arrow(< -r-0.2,    0.0,    0  >, <  r+0.2,    0.0,    0.0 >, at, Gray)
+arrow(<    0.0,    0.0, -r-0.2>, <    0.0,    0.0,  r+0.2 >, at, Gray)
+arrow(<    0.0, -r-0.2,    0  >, <    0.0,  r+0.2,    0.0 >, at, Gray)
+
+#declare gruen1  = rgb<0.0,0.4,0.0>;
+#declare gruen2  = rgb<0.0,0.4,0.8>;
+#declare orange1 = rgb<1.0,0.6,0.0>;
+#declare orange2 = rgb<0.8,0.0,0.4>;
+
+#macro ebene(v1, v2, farbe)
+	intersection {
+		box { <-r,-r,-r>, <r,r,r> }
+		plane { vnormalize(vcross(v1, v2)), 0.004 }
+		plane { vnormalize(-vcross(v1, v2)), 0.004 }
+		pigment {
+			color rgbt<farbe.x, farbe.y, farbe.z, 0.5>
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro gerade(v1, farbe)
+	intersection {
+		box { <-r,-r,-r>, <r,r,r> }
+		cylinder { -2 * r * vnormalize(v1),
+				2 * r * vnormalize(v1), 0.80*at }
+		pigment {
+			color farbe
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
+#macro kasten() 
+	difference {
+		box { <-r-0.01,-r-0.01,-r-0.01>, <r+0.01,r+0.01,r+0.01> }
+		union {
+			box {	<  -r,        -r,        -r     >,
+				<   r,         r,         r     >	}
+			box {	<-2*r,        -r+0.03,   -r+0.03>,
+				< 2*r,         r-0.03,    r-0.03>	}
+			box {	<  -r+0.03, -2*r,        -r+0.03>,
+				<   r-0.03,  2*r,         r-0.03>	}
+			box {	<  -r+0.03,   -r+0.03, -2*r     >,
+				<   r-0.03,    r-0.03,  2*r     >	}
+		}
+		pigment {
+			color rgb<0.8,0.8,0.8>
+		}
+		finish {
+			specular 0.9
+			metallic
+		}
+	}
+#end
+
diff --git a/vorlesungen/slides/5/beispiele/drei.jpg b/vorlesungen/slides/5/beispiele/drei.jpg
new file mode 100644
index 0000000..35f9034
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/drei.jpg
diff --git a/vorlesungen/slides/5/beispiele/drei.pov b/vorlesungen/slides/5/beispiele/drei.pov
new file mode 100644
index 0000000..bdc9630
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/drei.pov
@@ -0,0 +1,22 @@
+//
+// drei.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j21, at, orange2)
+//arrow(O, k21, at, gruen2)
+//arrow(O, k22, at, gruen2)
+gerade(j21, orange2)
+//ebene(k21, k22, gruen2)
+
+#declare at = 0.7 * at;
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+arrow(O, k11, at, gruen1)
+ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/kern1.jpg b/vorlesungen/slides/5/beispiele/kern1.jpg
new file mode 100644
index 0000000..5c99664
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern1.jpg
diff --git a/vorlesungen/slides/5/beispiele/kern1.pov b/vorlesungen/slides/5/beispiele/kern1.pov
new file mode 100644
index 0000000..8e61d8d
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern1.pov
@@ -0,0 +1,12 @@
+//
+// kern1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, k11, at, gruen1)
+gerade(k11, gruen1)
+
diff --git a/vorlesungen/slides/5/beispiele/kern2.jpg b/vorlesungen/slides/5/beispiele/kern2.jpg
new file mode 100644
index 0000000..87d18ac
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern2.jpg
diff --git a/vorlesungen/slides/5/beispiele/kern2.pov b/vorlesungen/slides/5/beispiele/kern2.pov
new file mode 100644
index 0000000..70127a2
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kern2.pov
@@ -0,0 +1,17 @@
+//
+// kern2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, k21, at, gruen2)
+arrow(O, k22, at, gruen2)
+ebene(k21, k22, gruen2)
+
+#declare at = 0.7 * at;
+arrow(O, k11, at, gruen1)
+gerade(k11, gruen1)
+
diff --git a/vorlesungen/slides/5/beispiele/kernbild.m b/vorlesungen/slides/5/beispiele/kernbild.m
new file mode 100644
index 0000000..28cd552
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild.m
@@ -0,0 +1,79 @@
+#
+# kernbild.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+rand("seed", 1291)
+rand("seed", 4711)
+
+lambda1 = 1;
+lambda2 = 1.8;
+
+A = [
+	lambda1,  0,  0;
+	0,  lambda2,  1;
+	0,  0, lambda2
+];
+
+B = eye(3) + rand(3,3);
+det(B)
+
+
+C = B*A*inverse(B)
+rank(C)
+
+# Eigenwert lambda1
+E2 = C - lambda1 * eye(3)
+rref(E2)
+
+# Eigenwert lambda2, k = 1
+E1 = C - lambda2 * eye(3)
+D = rref(E1);
+K1 = [
+	-D(1,3);
+	-D(2,3);
+	1
+];
+K1(:,1) = K1(:,1) / norm(K1(:,1));
+K1
+
+f = fopen("JK.inc", "w");
+fprintf(f, "//\n// JK.inc\n//\n// (c) 2021 Prof Dr Andreas Müller\n//\n\n");
+fprintf(f, "// Kern und Bild von C - %.3f I\n", lambda2);
+fprintf(f, "#declare k11 = < %.5f, %.5f, %.5f>;\n", K1(1,1), K1(2,1), K1(3,1));
+fprintf(f, "#declare j11 = < %.5f, %.5f, %.5f>;\n", E1(1,1), E1(2,1), E1(3,1));
+fprintf(f, "#declare j12 = < %.5f, %.5f, %.5f>;\n", E1(1,2), E1(2,2), E1(3,2));
+fprintf(f, "\n");
+
+# k = 2
+E12 = E1 * E1
+D = rref(E12);
+K2 = [
+	-D(1,2), -D(1,3);
+	      1,       0;
+	      0,       1
+]
+K2(:,1) = K2(:,1) / norm(K2(:,1));
+K2(:,2) = K2(:,2) / norm(K2(:,2));
+K2
+
+fprintf(f, "// Kern und Bild von (C - %.3f I)^2\n", lambda2);
+fprintf(f, "#declare k21 = < %.5f, %.5f, %.5f>;\n", K2(1,1), K2(2,1), K2(3,1));
+fprintf(f, "#declare k22 = < %.5f, %.5f, %.5f>;\n", K2(1,2), K2(2,2), K2(3,2));
+fprintf(f, "#declare j21 = < %.5f, %.5f, %.5f>;\n", E12(1,1), E12(2,1), E12(3,1));
+fprintf(f, "\n");
+
+fclose(f);
+
+# Verifikation
+x = K2 \ K1
+K2 * x
+
+eig(C)
+
+[U, S, V] = svd(C)
+
+
+s = rand("seed")
+
diff --git a/vorlesungen/slides/5/beispiele/kernbild1.jpg b/vorlesungen/slides/5/beispiele/kernbild1.jpg
new file mode 100644
index 0000000..87e874e
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild1.jpg
diff --git a/vorlesungen/slides/5/beispiele/kernbild1.pov b/vorlesungen/slides/5/beispiele/kernbild1.pov
new file mode 100644
index 0000000..425f299
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild1.pov
@@ -0,0 +1,15 @@
+//
+// kernbild1.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+arrow(O, k11, at, gruen1)
+ebene(j11, j12, orange1)
+
+//kasten()
diff --git a/vorlesungen/slides/5/beispiele/kernbild2.jpg b/vorlesungen/slides/5/beispiele/kernbild2.jpg
new file mode 100644
index 0000000..1160b31
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild2.jpg
diff --git a/vorlesungen/slides/5/beispiele/kernbild2.pov b/vorlesungen/slides/5/beispiele/kernbild2.pov
new file mode 100644
index 0000000..ae67ea1
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kernbild2.pov
@@ -0,0 +1,21 @@
+//
+// kernbild2.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j21, at, orange2)
+arrow(O, k21, at, gruen2)
+arrow(O, k22, at, gruen2)
+gerade(j21, orange2)
+ebene(k21, k22, gruen2)
+
+//arrow(O, j11, at, orange1)
+//arrow(O, j12, at, orange1)
+//arrow(O, k11, at, gruen1)
+//gerade(k11, gruen1)
+//ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/kombiniert.jpg b/vorlesungen/slides/5/beispiele/kombiniert.jpg
new file mode 100644
index 0000000..9cb789c
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kombiniert.jpg
diff --git a/vorlesungen/slides/5/beispiele/kombiniert.pov b/vorlesungen/slides/5/beispiele/kombiniert.pov
new file mode 100644
index 0000000..c187d08
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/kombiniert.pov
@@ -0,0 +1,22 @@
+//
+// kombiniert.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
+arrow(O, j21, at, orange2)
+arrow(O, k21, at, gruen2)
+arrow(O, k22, at, gruen2)
+gerade(j21, orange2)
+ebene(k21, k22, gruen2)
+
+#declare at = 0.7 * at;
+
+arrow(O, j11, at, orange1)
+arrow(O, j12, at, orange1)
+arrow(O, k11, at, gruen1)
+ebene(j11, j12, orange1)
+
diff --git a/vorlesungen/slides/5/beispiele/leer.jpg b/vorlesungen/slides/5/beispiele/leer.jpg
new file mode 100644
index 0000000..9789887
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/leer.jpg
diff --git a/vorlesungen/slides/5/beispiele/leer.pov b/vorlesungen/slides/5/beispiele/leer.pov
new file mode 100644
index 0000000..f4653d9
--- /dev/null
+++ b/vorlesungen/slides/5/beispiele/leer.pov
@@ -0,0 +1,9 @@
+//
+// leer.pov
+//
+// (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+//
+
+#include "common.inc"
+#include "JK.inc"
+
diff --git a/vorlesungen/slides/5/bloecke.tex b/vorlesungen/slides/5/bloecke.tex
new file mode 100644
index 0000000..974f238
--- /dev/null
+++ b/vorlesungen/slides/5/bloecke.tex
@@ -0,0 +1,141 @@
+%
+% bloecke.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\sx{1}
+\def\sy{0.1}
+\def\block#1#2{
+	\fill[color=red] ({#1},{-#1}) rectangle ({#1+#2},{-#1-#2});
+}
+\def\kreuz#1{
+	\draw[color=white,line width=0.1pt] (0,{-#1})--(60,{-#1});
+	\draw[color=white,line width=0.1pt] (#1,0)--(#1,-60);
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Blockgrössen aus $\dim\mathcal{K}^k(A)$ ablesen}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.56\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\coordinate (A) at ({1*\sx},{20*\sy});
+\coordinate (B) at ({2*\sx},{(20+15)*\sy});
+\coordinate (C) at ({3*\sx},{(20+15+10)*\sy});
+\coordinate (D) at ({4*\sx},{(20+15+10+8)*\sy});
+\coordinate (E) at ({5*\sx},{(20+15+10+8+5)*\sy});
+\coordinate (F) at ({6*\sx},{(20+15+10+8+5+2)*\sy});
+\fill[color=darkgreen!20] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F)
+	-- ({6*\sx},0) -- cycle;
+
+\fill[color=darkgreen!40] (0,0) -- ({1*\sx},0) -- (A) -- cycle;
+\fill[color=darkgreen!40] (A) -- ({2*\sx},{20*\sy}) -- (B) -- cycle;
+\fill[color=darkgreen!40] (B) -- ({3*\sx},{(20+15)*\sy}) -- (C) -- cycle;
+\fill[color=darkgreen!40] (C) -- ({4*\sx},{(20+15+10)*\sy}) -- (D) -- cycle;
+\fill[color=darkgreen!40] (D) -- ({5*\sx},{(20+15+10+8)*\sy}) -- (E) -- cycle;
+\fill[color=darkgreen!40] (E) -- ({6*\sx},{(20+15+10+8+5)*\sy}) -- (F) -- cycle;
+
+\draw[color=darkgreen,line width=1.4pt] (0,0) -- (A) -- (B) -- (C) -- (D) -- (E) -- (F);
+
+\draw[color=gray] (A) -- (0,{20*\sy});
+\draw[color=gray] (B) -- (0,{(20+15)*\sy});
+\draw[color=gray] (C) -- (0,{(20+15+10)*\sy});
+\draw[color=gray] (D) -- (0,{(20+15+10+8)*\sy});
+\draw[color=gray] (E) -- (0,{(20+15+10+8+5)*\sy});
+\draw[color=gray] (F) -- (0,{(20+15+10+8+5+2)*\sy});
+
+\node at ({0.5*\sx},{0.5*20*\sy})
+	[right] {$d_1 = \dim\mathcal{K}^1(A)-\dim\mathcal{K}^0(A)$};
+\node at ({1.5*\sx},{0.5*(20+20+15)*\sy})
+	[right] {$d_2 = \dim\mathcal{K}^2(A)-\dim\mathcal{K}^1(A)$};
+\node at ({2.5*\sx},{0.5*(2*20+2*15+1*10)*\sy}) [right] {$d_3$};
+\node at ({3.5*\sx},{0.5*(2*20+2*15+2*10+8)*\sy}) [right] {$d_4$};
+\node at ({4.5*\sx-0.1},{0.5*(2*20+2*15+2*10+2*8+5)*\sy+0.2}) [below right] {$d_5$};
+\node at ({5.5*\sx},{0.5*(2*20+2*15+2*10+2*8+2*5+2)*\sy+0.1}) [below] {$d_6$};
+
+\fill (A) circle[radius=0.08];
+\fill (B) circle[radius=0.08];
+\fill (C) circle[radius=0.08];
+\fill (D) circle[radius=0.08];
+\fill (E) circle[radius=0.08];
+\fill (F) circle[radius=0.08];
+
+\draw[->] (-0.1,0) -- ({6*\sx+1},0) coordinate[label={$k$}];
+\draw[->] (0,-0.1) -- (0,6.5) coordinate[label={right:$\dim\mathcal{K}^k(A)$}];
+
+\foreach \x in {0,1,...,6}{
+	\draw ({\sx*\x},{-0.05}) -- ({\sx*\x},0.05);
+	\node at ({\sx*\x},{-0.1}) [below] {$\x$};
+}
+
+\node at (0,{60*\sy}) [left] {\llap{$n$}};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.43\textwidth}
+\vspace{-10pt}
+\begin{center}
+\begin{tabular}{>{$}c<{$}|>{$}r<{$}|>{$}c<{$}|>{$}c<{$}}
+k&d_k&\# M_k(\Bbbk)\text{-Blöcke}&\text{Beispiel}\\
+\hline
+0&  0&        &\\
+1& 20& d_1-d_2&5\\
+2& 15& d_2-d_3&5\\
+3& 10& d_3-d_4&2\\
+4&  8& d_4-d_5&3\\
+5&  5& d_5-d_6&3\\
+6&  2& d_6    &2\\
+\end{tabular}
+\end{center}
+\vspace{-13pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.05]
+\fill[color=gray!40] (0,0) rectangle (60,-60);
+\node[color=white] at (30,-30) [scale=6] {$A$};
+\kreuz{5}
+\kreuz{15}
+\kreuz{21}
+\kreuz{33}
+\kreuz{48}
+\node at (0,-2.5) [left] {$k=1$};
+\node at (60,-2.5) [right] {$5$ Blöcke};
+\node at (0,-10) [left] {$k=2$};
+\node at (60,-10) [right] {$5$ Blöcke};
+\node at (0,-18) [left] {$k=3$};
+\node at (60,-18) [right] {$2$ Blöcke};
+\node at (0,-27) [left] {$k=4$};
+\node at (60,-27) [right] {$3$ Blöcke};
+\node at (0,-40.5) [left] {$k=5$};
+\node at (60,-40.5) [right] {$3$ Blöcke};
+\node at (0,-54) [left] {$k=6$};
+\node at (60,-54) [right] {$2$ Blöcke};
+\block{0}{1}
+\block{1}{1}
+\block{2}{1}
+\block{3}{1}
+\block{4}{1}
+\block{5}{2}
+\block{7}{2}
+\block{9}{2}
+\block{11}{2}
+\block{13}{2}
+\block{15}{3}
+\block{18}{3}
+\block{21}{4}
+\block{25}{4}
+\block{29}{4}
+\block{33}{5}
+\block{38}{5}
+\block{43}{5}
+\block{48}{6}
+\block{54}{6}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/cayleyhamilton.tex b/vorlesungen/slides/5/cayleyhamilton.tex
new file mode 100644
index 0000000..c0813be
--- /dev/null
+++ b/vorlesungen/slides/5/cayleyhamilton.tex
@@ -0,0 +1,91 @@
+%
+% cayleyhamilton.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Satz von Cayley-Hamilton}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Ein Eigenwert $\lambda$\strut}
+$A$ besteht aus
+$b$ Blöcken $J_\lambda$ mit maximaler Dimension $l$:
+\phantom{blubb\strut}
+\begin{align*}
+\uncover<2->{
+\chi_{A}(X)
+&=
+\det (A-XI) = (\lambda-X)^n
+}
+\\
+\uncover<3->{
+m_{A}(X)
+&=
+(\lambda-X)^l
+}
+\\
+\uncover<4->{
+b&= \ker A
+}
+\end{align*}
+\uncover<5->{%
+Wegen $l \le n$ folgt
+\[
+m_A(X) | \chi_A(X)
+\uncover<6->{\quad\Rightarrow\quad
+\chi_A(A) = 0}
+\]}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<7->{%
+\begin{block}{$A=A_1\oplus\dots\oplus A_k$}
+\uncover<8->{%
+$A_i\in M_{n_i}(\Bbbk)$ mit EW $\lambda_i$,
+$A_i$ besteht aus
+$b_i$ Blöcken $J_{\lambda_i}$ mit max.~Dimension $l_i$\strut:}
+\begin{align*}
+\uncover<9->{
+\chi_A(X)
+&=
+(\lambda_1-X)^{n_1}
+\dots
+(\lambda_k-X)^{n_k}
+}
+\\
+\uncover<10->{
+m_A(X)
+&=
+(\lambda_1-X)^{l_1}
+\dots
+(\lambda_k-X)^{l_k}
+}
+\\
+\uncover<11->{
+b_i &= \ker (A-\lambda_iI)
+}
+\end{align*}
+\uncover<12->{%
+$A=A_1\oplus\dots\oplus A_k$}
+\begin{align*}
+\uncover<13->{
+\chi_{A_i}(A_i)&=0\;\forall i
+}
+\\
+\uncover<14->{%
+\chi_A(A) &=
+\chi_{A_1}(A)\dots\chi_{A_k}(A)
+ = 0}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<15->{%
+\begin{block}{Satz}
+Für jede Matrix $A\in M_n(\Bbbk)$ gilt
+$m_A(X) | \chi_A(X)$ oder $\chi_A(A)=0$
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/chapter.tex b/vorlesungen/slides/5/chapter.tex
new file mode 100644
index 0000000..96eea29
--- /dev/null
+++ b/vorlesungen/slides/5/chapter.tex
@@ -0,0 +1,36 @@
+%
+% chapter.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswi
+%
+\folie{5/verzerrung.tex}
+\folie{5/motivation.tex}
+\folie{5/charpoly.tex}
+\folie{5/kernbildintro.tex}
+\folie{5/kernbilder.tex}
+\folie{5/kernbild.tex}
+\folie{5/ketten.tex}
+\folie{5/dimension.tex}
+\folie{5/folgerungen.tex}
+\folie{5/injektiv.tex}
+\folie{5/nilpotent.tex}
+\folie{5/eigenraeume.tex}
+\folie{5/zerlegung.tex}
+\folie{5/normalnilp.tex}
+\folie{5/bloecke.tex}
+\folie{5/jordanblock.tex}
+\folie{5/jordan.tex}
+\folie{5/reellenormalform.tex}
+\folie{5/cayleyhamilton.tex}
+\folie{5/konvergenzradius.tex}
+\folie{5/krbeispiele.tex}
+\folie{5/spektralgelfand.tex}
+\folie{5/Aiteration.tex}
+\folie{5/satzvongelfand.tex}
+\folie{5/stoneweierstrass.tex}
+\folie{5/potenzreihenmethode.tex}
+\folie{5/logarithmusreihe.tex}
+\folie{5/exponentialfunktion.tex}
+\folie{5/hyperbolisch.tex}
+\folie{5/spektrum.tex}
+\folie{5/normal.tex}
diff --git a/vorlesungen/slides/5/charpoly.tex b/vorlesungen/slides/5/charpoly.tex
new file mode 100644
index 0000000..63bfee5
--- /dev/null
+++ b/vorlesungen/slides/5/charpoly.tex
@@ -0,0 +1,78 @@
+%
+% charpoly.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Charakteristisches Polynom über $\mathbb{C}$}
+\vspace{-18pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenwerte}
+Nur diejenigen $\mu$ kommen in Frage, für die
+$A-\mu I$ singulär ist:
+\[
+\chi_{A}(\mu)
+=
+\det (A-\mu I) = 0
+\]
+$\Rightarrow$ $\mu$ ist Nullstelle von $\chi_{A}(X)\in\mathbb{C}[X]$
+\end{block}
+\uncover<2->{%
+\begin{block}{Zerlegung in Linearfaktoren}
+$\mu_1,\dots,\mu_n$ die Nullstellen von $\chi_A(X)$:
+\[
+\chi_A(X)
+=
+(X-\mu_1)\dots (X-\mu_n)
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Fundamentalsatz der Algebra}
+Über $\mathbb{C}$ zerfällt jedes Polynom in $\mathbb{C}[X]$ in
+Linearfaktoren
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Minimalpolynom}
+Alle Nullstellen von $\chi_A(X)$ müssen in $m_A(X)$ vorkommen
+\end{block}}
+\uncover<5->{%
+\begin{proof}[Beweis]
+\begin{enumerate}
+\item<6->
+$m_A(X) = (X-\lambda) \prod_{i\in I}(X-\mu_i)$
+\item<7->
+$A-\lambda I$ ist regulär
+\end{enumerate}
+\uncover<8->{%
+\begin{align*}
+&\Rightarrow&
+m_A(A)&=0
+\\
+&&
+\uncover<9->{
+(A-\lambda)^{-1}m_A(A) &=0
+}
+\\
+&&
+\uncover<10->{
+\prod_{i\in I}(A-\mu_i)&=0,
+}
+\end{align*}}
+\uncover<11->{%
+d.~h.~\(
+\displaystyle
+\overline{m}_A(X)
+=
+\prod_i{i\in I}(X-\mu_i)
+\in
+\mathbb{C}[X]
+\)}
+\end{proof}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/dimension.tex b/vorlesungen/slides/5/dimension.tex
new file mode 100644
index 0000000..ff687b3
--- /dev/null
+++ b/vorlesungen/slides/5/dimension.tex
@@ -0,0 +1,68 @@
+%
+% dimension.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Dimension von $\mathcal{K}^k(f)$ und $\mathcal{J}^k(f)$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\def\pfad{
+	(0,0) -- (1,0.3) -- (2,0.9) 
+	--
+	(4,2.4) -- (5,2.7) -- (6,3.3)
+	--
+	(8,3.7) -- (9,4) -- (10,4) -- (11,4) -- (12,4)
+}
+
+\fill[color=darkgreen!20] \pfad -- (12,0) -- cycle;
+\fill[color=orange!20] \pfad -- (12,6) -- (0,6) -- cycle;
+
+\fill[color=darkgreen!40] (9,0) -- (12,0) -- (12,4) -- (9,4) -- cycle;
+\fill[color=orange!40] (9,4) -- (12,4) -- (12,6) -- (9,6) -- cycle;
+
+\node[color=orange] at (10.5,5) {$\mathcal{J}(f)$};
+\node[color=darkgreen] at (10.5,2) {$\mathcal{K}(f)$};
+
+\node[color=orange] at (5.5,4.5) {$\mathcal{J}^k(f)\supset\mathcal{J}^{k+1}(f)$};
+\node[color=darkgreen] at (5.5,1.5) {$\mathcal{K}^k(f)\subset\mathcal{K}^{k+1}(f)$};
+
+\draw[line width=1.4pt] \pfad;
+
+\draw[->] (-0.1,6) -- (12.5,6) coordinate[label={$k$}];
+\draw[->] (-0.1,0) -- (12.5,0) coordinate[label={$k$}];
+\node at (-0.1,6) [left] {$n$};
+\node at (-0.1,0) [left] {$0$};
+\foreach \x in {0,1,2,4,5,6,8,9,10,11,12}{
+	\fill (\x,0) circle[radius=0.05];
+	\fill (\x,6) circle[radius=0.05];
+}
+\node at (0,0) [below] {$0$};
+\node at (1,0) [below] {$1$};
+\node at (2,0) [below] {$2$};
+
+\node at (4,0) [below] {$k-1$};
+\node at (5,0) [below] {$k$};
+\node at (6,0) [below] {$k+1$};
+
+\node at (8,0) [below] {$l-1$};
+\node at (9,0) [below] {$l$};
+\node at (10,0) [below] {$l+1$};
+\node at (11,0) [below] {$l+2$};
+\node at (12,0) [below] {$l+3$};
+
+\fill (9,4) circle[radius=0.05];
+
+\node[color=orange] at (-0.2,3) [rotate=90] {$\dim\mathcal{J}^k(f)$};
+\node[color=darkgreen] at (12.2,2) [rotate=-90] {$\dim\mathcal{K}^k(f)$};
+
+\node[color=orange] at (9,5) [rotate=-90] {$\dim\mathcal{J}(f)$};
+\node[color=darkgreen] at (9,2) [rotate=-90] {$\dim\mathcal{K}(f)$};
+
+\end{tikzpicture}
+\end{center}
+
+\end{frame}
diff --git a/vorlesungen/slides/5/eigenraeume.tex b/vorlesungen/slides/5/eigenraeume.tex
new file mode 100644
index 0000000..fd4803c
--- /dev/null
+++ b/vorlesungen/slides/5/eigenraeume.tex
@@ -0,0 +1,48 @@
+%
+% eigenraeume.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Eigenräume}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenraum}
+Für $\lambda\in\Bbbk$ heisst
+\begin{align*}
+E_\lambda(f)
+&=
+\ker (f-\lambda)
+\\
+\uncover<2->{
+&=
+\{v\in V\;|\; f(v) = \lambda v\}
+}
+\end{align*}
+\uncover<3->{%
+{\em Eigenraum} von $f$ zum Eigenwert $\lambda$.}
+\end{block}
+\uncover<4->{%
+$E_\lambda(f)\subset V$ ist ein Unterraum}
+
+\uncover<5->{%
+\begin{block}{Eigenwert}
+Falls $\dim E_\lambda(f)>0$ heisst $\lambda$ Eigenwert von $f$.
+\end{block}}
+
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{verallgemeinerter Eigenraum}
+Für $\lambda\in \Bbbk$ heisst
+\[
+\mathcal{E}_\lambda(f)
+=
+\mathcal{K}(f-\lambda)
+\]
+verallgemeinerter Eigenraum
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/exponentialfunktion.tex b/vorlesungen/slides/5/exponentialfunktion.tex
new file mode 100644
index 0000000..caae16b
--- /dev/null
+++ b/vorlesungen/slides/5/exponentialfunktion.tex
@@ -0,0 +1,131 @@
+%
+% exponentialfunktion.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Exponentialfunktion}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\only<1-6>{%
+\ifthenelse{\boolean{presentation}}{
+\begin{column}{0.48\textwidth}
+\begin{block}{$x(t) \in\mathbb{R}$}
+\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}x(t) &= ax(t) &a&\in\mathbb{R}
+\\
+x(0) &= c&&\in\mathbb{R}
+\intertext{\uncover<2->{Lösung:}}
+\uncover<2->{x(t) &= ce^{at}}
+\end{align*}
+\end{block}
+\end{column}}{}}
+\begin{column}{0.48\textwidth}
+\uncover<3->{%
+\begin{block}{$X(t) \in M_n(\mathbb{R})$}
+\vspace{-10pt}
+\begin{align*}
+\frac{d}{dt}X(t)
+&=
+A
+X(t)&A&\in M_n(\mathbb{R})
+\\
+X(0)&=C&&\in M_n(\mathbb{R})
+\intertext{\uncover<4->{gekoppelte Differentialgleichung für
+vier Funktionen $x_{ij}(t)$}}
+\uncover<5->{\dot{x}_{11} &= \rlap{$a_{11} x_{11}(t) + a_{12} x_{21}(t)$}}\\
+\uncover<5->{\dot{x}_{12} &= \rlap{$a_{11} x_{12}(t) + a_{12} x_{22}(t)$}}\\
+\uncover<5->{\dot{x}_{21} &= \rlap{$a_{21} x_{11}(t) + a_{22} x_{21}(t)$}}\\
+\uncover<5->{\dot{x}_{22} &= \rlap{$a_{21} x_{12}(t) + a_{22} x_{22}(t)$}}\\
+\intertext{\uncover<6->{Lösung:}}
+\uncover<6->{X(t) &= \exp(At) C}
+\end{align*}
+\end{block}}
+\end{column}
+\only<7-9>{%
+\ifthenelse{\boolean{presentation}}{
+\begin{column}{0.48\textwidth}
+\begin{block}{Beispiel: Diagonalmatrix}
+%$D=\operatorname{diag}(\lambda_1,\dots,\lambda_n)$
+\begin{align*}
+\frac{d}{dt}X&=DX &&\uncover<8->{\Rightarrow &\dot{x}_{ij}(t) &= \lambda_i x_{ij}(t)}
+\\
+X(0)&=C
+&&\uncover<8->{\Rightarrow&x_{ij}(t)&=c_{ij}}
+\end{align*}
+\uncover<9->{%
+Lösung:
+\[
+x_{ij}(t) =c_{ij}e^{\lambda_i t}
+\]}
+\end{block}
+\end{column}}{}}
+\uncover<10->{%
+\begin{column}{0.48\textwidth}
+\begin{block}{Beispiel: Jordan-Block}
+\vspace{-10pt}
+\begin{align*}
+A&=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}
+\rlap{$\displaystyle,\;
+X(t)
+=
+\ifthenelse{\boolean{presentation}}{
+\only<22>{
+	e^{\lambda t}
+	\begin{pmatrix} 1&t/\lambda\\ 0&1 \end{pmatrix}
+}}{}
+\only<23>{
+	\frac{e^{\lambda t}}{\lambda}
+	\begin{pmatrix} \lambda&t\\ 0&\lambda \end{pmatrix}
+}
+C
+$}
+\\
+\uncover<11->{
+\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + \phantom{\lambda}x_{2i}(t),&&x_{1i}(0)&=c_{1i}
+}
+\\
+\uncover<12->{
+\dot{x}_{2i}(t)&=\phantom{\lambda x_{1i}(t)+\mathstrut}\lambda x_{2i}(t),&&x_{2i}(0)&=c_{2i}
+}
+\end{align*}
+\uncover<13->{%
+Lösung:}
+\begin{align*}
+\uncover<14->{
+x_{2i}(t)&=c_{2i}e^{\lambda t}
+}
+\\
+\uncover<15->{
+\dot{x}_{1i}(t)&=\lambda x_{1i}(t) + c_{2i}e^{\lambda t}
+}
+\\
+\ifthenelse{\boolean{presentation}}{
+\only<16-17>{x_{1i\only<16>{,h}}(t)}}{}
+\only<18->{\dot{x}_{1i}(t)}
+&
+\ifthenelse{\boolean{presentation}}{
+\only<16-17>{=c\only<17>{(t)}\lambda e^{\lambda t}}
+\only<18>{=\dot{c}(t)\lambda e^{\lambda t}
++
+c(t)\lambda^2 e^{\lambda t}}
+}{}
+\only<19->{=\lambda x_{1i}(t) + \dot{c}(t)\lambda e^{\lambda t}}
+\\
+\uncover<20->{\Rightarrow
+\dot{c}(t)&= c_{2i}/\lambda
+\Rightarrow
+c(t) = c_{2i}(0) +tc_{2i}/\lambda
+}
+\\
+\uncover<21->{
+x_{1i}(t) & =c_{1i}e^{\lambda t} + t(c_{2i}/\lambda)e^{\lambda t}
+}
+\end{align*}
+\end{block}
+\end{column}}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/folgerungen.tex b/vorlesungen/slides/5/folgerungen.tex
new file mode 100644
index 0000000..4a8dbe6
--- /dev/null
+++ b/vorlesungen/slides/5/folgerungen.tex
@@ -0,0 +1,84 @@
+%
+% folgerungen.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\sx{1}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Folgerungen}
+\vspace{-10pt}
+\begin{columns}[t]
+\begin{column}{0.30\textwidth}
+\begin{block}{Zunahme}
+Für alle $k<l$ gilt
+\begin{align*}
+\mathcal{J}^k(f) &\supsetneq \mathcal{J}^{k+1}(f)
+\\
+\mathcal{K}^k(f) &\subsetneq \mathcal{K}^{k+1}(f)
+\end{align*}
+Für $k\ge l$ gilt
+\begin{align*}
+\mathcal{J}^k(f) &= \mathcal{J}^{k+1}(f)
+\\
+\mathcal{K}^k(f) &= \mathcal{K}^{k+1}(f)
+\end{align*}
+Ausserdem ist $l\le n$
+\end{block}
+\end{column}
+\begin{column}{0.66\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\pfad{
+	({0*\sx},6) --
+	({1*\sx},4.5) --
+	({2*\sx},3.5) --
+	({3*\sx},2.9) --
+	({4*\sx},2.6) --
+	({5*\sx},2.4) --
+	({6*\sx},2.4)
+}
+
+\fill[color=orange!20] \pfad -- ({6*\sx},0) -- (0,0) -- cycle;
+\fill[color=darkgreen!20] \pfad -- ({6*\sx},6) -- cycle;
+\fill[color=orange!40] ({5*\sx},0) rectangle ({6*\sx},2.4);
+\fill[color=darkgreen!40] ({5*\sx},6) rectangle ({6*\sx},2.4);
+
+\draw[color=darkgreen,line width=2pt] ({3*\sx},6) -- ({3*\sx},2.9);
+\node[color=darkgreen] at ({3*\sx},4.45) [rotate=90,above] {$\dim\mathcal{K}^k(A)$};
+\draw[color=orange,line width=2pt] ({3*\sx},0) -- ({3*\sx},2.9);
+\node[color=orange] at ({3*\sx},1.45) [rotate=90,above] {$\dim\mathcal{J}^k(A)$};
+
+\node[color=orange] at ({5.5*\sx},1.2) [rotate=90] {bijektiv};
+\node[color=darkgreen] at ({5.5*\sx},4.2) [rotate=90] {konstant};
+
+\fill ({0*\sx},6) circle[radius=0.08];
+\fill ({1*\sx},4.5) circle[radius=0.08];
+\fill ({2*\sx},3.5) circle[radius=0.08];
+\fill ({3*\sx},2.9) circle[radius=0.08];
+\fill ({4*\sx},2.6) circle[radius=0.08];
+\fill ({5*\sx},2.4) circle[radius=0.08];
+\fill ({6*\sx},2.4) circle[radius=0.08];
+
+\draw \pfad;
+
+\draw[->] (-0.1,0) -- ({6*\sx+0.5},0) coordinate[label={$k$}];
+\draw[->] (-0.1,6) -- ({6*\sx+0.5},6);
+
+\foreach \x in {0,...,6}{
+	\draw (\x,-0.05) -- (\x,0.05);
+}
+\foreach \x in {0,...,3}{
+	\node at ({\x*\sx},-0.05) [below] {$\x$};
+}
+\node at ({4*\sx},-0.05) [below] {$\dots\mathstrut$};
+\node at ({5*\sx},-0.05) [below] {$l$};
+\node at ({6*\sx},-0.05) [below] {$l+1$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/hyperbolisch.tex b/vorlesungen/slides/5/hyperbolisch.tex
new file mode 100644
index 0000000..905082a
--- /dev/null
+++ b/vorlesungen/slides/5/hyperbolisch.tex
@@ -0,0 +1,105 @@
+%
+% hyperbolisch.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Hyperbolische Funktionen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Differentialgleichung}
+\vspace{-10pt}
+\begin{align*}
+\ddot{y} &= y
+\;\Rightarrow\;
+\frac{d}{dt}
+\begin{pmatrix}y\\y_1\end{pmatrix}
+=
+\begin{pmatrix}0&1\\1&0\end{pmatrix}
+\begin{pmatrix}y\\y_1\end{pmatrix}
+\\
+y(0)&=a,\qquad y'(0)=b
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Lösung}
+\vspace{-13pt}
+\begin{align*}
+\lambda^2-1&=0
+\uncover<3->{
+\qquad\Rightarrow\qquad \lambda=\pm 1
+}
+\\
+\uncover<4->{
+y(t)&=Ae^t+Be^{-t}}
+\uncover<5->{
+\Rightarrow
+\left\{
+\arraycolsep=1.4pt
+\begin{array}{rcrcr}
+A&+&B&=&a\\
+A&-&B&=&b
+\end{array}
+\right.}
+\\
+&\uncover<6->{
+=\frac{a+b}2e^t + \frac{a-b}2e^{-t}}
+\\
+&\uncover<7->{=
+a{\color{darkgreen}\frac{e^t+e^{-t}}2} + b{\color{red}\frac{e^t-e^{-t}}2}}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.49\textwidth}
+\uncover<8->{%
+\begin{block}{Potenzreihe}
+\vspace{-12pt}
+\begin{align*}
+K&=\begin{pmatrix}0&1\\1&0\end{pmatrix}
+\uncover<10->{\quad\Rightarrow\quad K^2=I}
+\\
+\uncover<9->{
+e^{Kt}
+&=
+I+K+\frac1{2!}K^2 + \frac{1}{3!}K^3 + \frac{1}{4!}K^4+\dots
+}
+\\
+\uncover<11->{
+&=
+\biggl( 1+\frac{t^2}{2!} + \frac{t^4}{4!}+\dots \biggr)I
+}
+\\
+\uncover<11->{
+&\qquad
++\biggl(t+\frac{t^3}{3!}+\frac{t^5}{5!}+\dots\biggr)K
+}
+\\
+\uncover<12->{
+&=
+I{\,\color{darkgreen}\cosh t}  + K{\,\color{red}\sinh t}
+}
+\\
+\uncover<13->{
+\begin{pmatrix}y(t)\\y_1(t)\end{pmatrix}
+&=
+e^{Kt}\begin{pmatrix}a\\b\end{pmatrix}
+}
+\uncover<14->{
+=
+\begin{pmatrix}
+a{\,\color{darkgreen}\cosh t} + b{\,\color{red}\sinh t}\\
+a{\,\color{red}\sinh t} + b{\,\color{darkgreen}\cosh t}
+\end{pmatrix}
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/injektiv.tex b/vorlesungen/slides/5/injektiv.tex
new file mode 100644
index 0000000..90cbcd6
--- /dev/null
+++ b/vorlesungen/slides/5/injektiv.tex
@@ -0,0 +1,81 @@
+%
+% injektiv.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\sx{1.05}
+\begin{frame}[t]
+\frametitle{$f$ injektiv auf $\mathcal{J}(f)$}
+\setlength{\abovedisplayskip}{8pt}
+\setlength{\belowdisplayskip}{8pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.58\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=orange!20]
+	({0*\sx},-3.0) -- ({1*\sx},-2.0) -- ({2*\sx},-1.5) --
+	({3*\sx},-1.1) -- ({4*\sx},-0.9) -- ({5*\sx},-0.8) --
+	({6*\sx},-0.8) --
+	({6*\sx},0.8) -- ({5*\sx},0.8) -- ({4*\sx},0.9) --
+	({3*\sx},1.1) -- ({2*\sx},1.5) -- ({1*\sx},2.0) --
+	({0*\sx},3.0) -- cycle;
+\fill[color=orange!40] (0,-0.8) rectangle ({6*\sx},0.8);
+
+\foreach \x in {0,...,6}{
+	\draw[color=gray,line width=3pt] ({\x*\sx},-3)--({\sx*\x},3);
+}
+\foreach \x in {0,1,2,3}{
+	\node at ({\sx*\x},-3) [below] {$\x$};
+}
+\node at ({\sx*5},-3) [below] {$l$};
+\node at ({\sx*6},-3) [below] {$l+1$};
+\draw[->] (-0.1,-3.5) -- ({6*\sx+0.4},-3.5) coordinate[label={below:$k$}];
+
+\draw[line width=3pt,color=orange] ({0*\sx},-3.0) -- ({0*\sx},3.0);
+\draw[line width=3pt,color=orange] ({1*\sx},-2.0) -- ({1*\sx},2.0);
+\draw[line width=3pt,color=orange] ({2*\sx},-1.5) -- ({2*\sx},1.5);
+\draw[line width=3pt,color=orange] ({3*\sx},-1.1) -- ({3*\sx},1.1);
+\draw[line width=3pt,color=orange] ({4*\sx},-0.9) -- ({4*\sx},0.9);
+\draw[line width=3pt,color=orange] ({5*\sx},-0.8) -- ({5*\sx},0.8);
+\draw[line width=3pt,color=orange] ({6*\sx},-0.8) -- ({6*\sx},0.8);
+
+\foreach \x in {0,1,2,3}{
+	\node at ({\x*\sx},0) [rotate=90] {$\mathcal{J}^{\x}(A)$};
+}
+\node at ({4*\sx},0) {$\cdots$};
+\node at ({5*\sx},0) [rotate=90] {$\mathcal{J}^{l}(A)$};
+\node at ({6*\sx},0) [rotate=90] {$\mathcal{J}^{l+1}(A)$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.38\textwidth}
+\begin{block}{stationär}
+$l$ der $k$-Wert, ab dem gilt
+\begin{align*}
+\mathcal{J}^l(A) &= \mathcal{J}^{l+1}(A) = A\mathcal{J}^l(A)
+\end{align*}
+\end{block}
+\vspace{-10pt}
+\uncover<2->{%
+\begin{block}{Dimension}
+\vspace{-10pt}
+\[
+\dim \mathcal{J}^l(A) = \dim\mathcal{J}^{l+1}(A)
+\]
+\uncover<3->{%
+d.~h.~$A$ ist bijektiv als Selbstabbildung von
+$\mathcal{J}(A)$}
+\uncover<4->{%
+\[
+\Downarrow
+\]
+$A|\mathcal{J}(A)$ ist injektiv}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/jordan.tex b/vorlesungen/slides/5/jordan.tex
new file mode 100644
index 0000000..e6ece47
--- /dev/null
+++ b/vorlesungen/slides/5/jordan.tex
@@ -0,0 +1,138 @@
+%
+% jordan.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\L#1{
+	\node at ({#1-0.5},{0.5-#1}) {$\lambda$};
+}
+\def\E#1{
+	\node at ({#1-0.5},{1.5-#1}) {$1$};
+}
+
+\begin{frame}[t]
+\frametitle{Jordan Normalform}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.40\textwidth}
+\begin{block}{Wahl der Basis}
+\begin{enumerate}
+\item<2-> Zerlegung in verallgemeinerte Eigenräume
+\begin{align*}
+V
+&=
+\mathcal{E}_{{\color{blue}\lambda}}(A)
+\oplus
+\mathcal{E}_{{\color{darkgreen}\lambda}}(A)
+\oplus
+\mathcal{E}_{{\color{red}\lambda}}(A)
+%\oplus
+%\dots
+\\
+\llap{$A\mathcal{E}_{{\color{blue}\lambda}}$}(A)
+&\subset
+\mathcal{E}_{{\color{blue}\lambda}}(A)
+\\
+\llap{$A\mathcal{E}_{{\color{darkgreen}\lambda}}$}(A)
+&\subset
+\mathcal{E}_{{\color{darkgreen}\lambda}}(A)
+\\
+\llap{$A\mathcal{E}_{{\color{red}\lambda}}$}(A)
+&\subset
+\mathcal{E}_{{\color{red}\lambda}}(A),
+\dots
+\end{align*}
+\item<3-> In jedem Eigenraum: Zerlegung in Jordan-Blöcke
+\end{enumerate}
+\end{block}
+\end{column}
+\begin{column}{0.56\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.33]
+\fill[color=gray!20] (0,-20) rectangle (20,0);
+\node[color=white] at (10,-10) [scale=12] {$A$};
+
+\uncover<2->{
+	\fill[color=blue!20,opacity=0.5] (0,0) rectangle (8,-8);
+	\fill[color=darkgreen!20,opacity=0.5] (8,-8) rectangle (15,-15);
+	\fill[color=red!20,opacity=0.5] (15,-15) rectangle (20,-20);
+	\fill[color=blue!20] (0,0) rectangle (8,2);
+	\fill[color=blue!20] (-2,-8) rectangle (0,0);
+	\fill[color=darkgreen!20] (8,0) rectangle (15,2);
+	\fill[color=darkgreen!20] (-2,-15) rectangle (0,-8);
+	\fill[color=red!20] (15,0) rectangle (20,2);
+	\fill[color=red!20] (-2,-20) rectangle (0,-15);
+}
+
+\uncover<3->{
+	\draw[color=gray] (0,0) rectangle (5,-5);
+	\draw[color=gray] (5,-5) rectangle (8,-8);
+	\draw[color=gray] (8,-8) rectangle (15,-15);
+	\draw[color=gray] (15,-15) rectangle (16,-16);
+	\draw[color=gray] (16,-16) rectangle (17,-17);
+	\draw[color=gray] (17,-17) rectangle (20,-20);
+}
+
+\uncover<2->{
+	\draw[color=gray] (8,0) -- (8,-20);
+	\draw[color=gray] (15,0) -- (15,-20);
+	\draw[color=gray] (0,-8) -- (20,-8);
+	\draw[color=gray] (0,-15) -- (20,-15);
+
+	\node at (0,-4) [above,rotate=90]
+		{$\mathcal{E}_{{\color{blue}\lambda}}(A)$};
+	\node at (4,0) [above]
+		{$\mathcal{E}_{{\color{blue}\lambda}}(A)$};
+	\node at (0,-11.5) [above,rotate=90]
+		{$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$};
+	\node at (11.5,0) [above]
+		{$\mathcal{E}_{{\color{darkgreen}\lambda}}(A)$};
+	\node at (0,-18.5) [above,rotate=90]
+		{$\mathcal{E}_{{\color{red}\lambda}}(A)$};
+	\node at (18.5,0) [above]
+		{$\mathcal{E}_{{\color{red}\lambda}}(A)$};
+}
+
+\uncover<2->{
+	{\color{blue}
+	\foreach \x in {1,...,8}{ \L{\x} }
+	}
+	{\color{darkgreen}
+	\foreach \x in {9,...,15}{ \L{\x} }
+	}
+	{\color{red}
+	\foreach \x in {16,...,20}{ \L{\x} }
+	}
+}
+
+\uncover<3->{
+\E{2}
+\E{3}
+\E{4}
+\E{5}
+
+\E{7}
+\E{8}
+
+\E{10}
+\E{11}
+\E{12}
+\E{13}
+\E{14}
+\E{15}
+
+\E{19}
+\E{20}
+}
+
+\draw (0,-20) rectangle (20,0);
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/5/jordanblock.tex b/vorlesungen/slides/5/jordanblock.tex
new file mode 100644
index 0000000..1c3bce9
--- /dev/null
+++ b/vorlesungen/slides/5/jordanblock.tex
@@ -0,0 +1,68 @@
+%
+% jordanblock.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\def\NL{
+\ifthenelse{\boolean{presentation}}{
+\only<-8>{\phantom{\lambda}\llap{$0$}}\only<9->{\lambda}
+}{
+\lambda
+}
+}
+
+\begin{frame}[t]
+\frametitle{Jordan-Block}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Gegeben}
+Matrix $A\in M_n(\Bbbk)$ derart, dass
+\begin{itemize}
+\item<2->
+$A-\lambda I$ nilpotent
+\item<5->
+$A^{n-1}\ne 0$
+\end{itemize}
+\end{block}
+\vspace{-5pt}
+\uncover<3->{
+\begin{block}{Folgerungen}
+Es gibt eine Basis derart, dass
+\begin{enumerate}
+\item<4->
+$A-\lambda I$ hat Normalform einer nilpotenten Matrix
+\item<6->
+Es gibt nur einen Block, da $\dim\ker(A-\lambda I)=1$
+\end{enumerate}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{\ifthenelse{\boolean{presentation}}{\only<-8>{Normalform einer nilpotenten Matrix\strut}}{}\only<9->{Normalform: genau ein Eigenwert\strut}}
+\[
+A\uncover<-8>{-\lambda I}=\begin{pmatrix}
+\NL &1& & & & & & & \\
+ &\NL &1& & & & & & \\
+ & &\NL &\uncover<7->{{\color<7>{red}1}}& & & & & \\
+ & & &\NL &1& & & & \\
+ & & & &\NL &1& & & \\
+ & & & & &\NL &1& & \\
+ & & & & & &\NL &\uncover<7->{{\color<7>{red}1}}& \\
+ & & & & & & &\NL &\uncover<7->{{\color<7>{red}1}}\\
+ & & & & & & & &\NL
+\end{pmatrix}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-5pt}
+\uncover<8->{%
+\begin{block}{Jordan-Normalform}
+In dieser Basis hat $A$ Jordan-Normalform
+\end{block}}
+\end{frame}
+
+\egroup
diff --git a/vorlesungen/slides/5/kernbild.tex b/vorlesungen/slides/5/kernbild.tex
new file mode 100644
index 0000000..3890717
--- /dev/null
+++ b/vorlesungen/slides/5/kernbild.tex
@@ -0,0 +1,86 @@
+%
+% kernbild.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Kern und Bild}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\uncover<1->{%
+\begin{block}{Kern}
+Lineare Abbildung $f\colon V\to V$
+\[
+\ker f = \mathcal{K}(F) = \{v\in V\;|\; f(v)=0\}
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Kern von $A^k$}
+\[
+\mathcal{K}^k(f) = \operatorname{ker} f^k
+\]
+\begin{align*}
+\uncover<5->{
+\mathcal{K}^k(f)
+&=
+\{v\in V\;|\; f^{k}(v)=0\}
+}
+\\
+\uncover<6->{
+&\subset
+\{v\in V\;|\; f^{k+1}(v)=0\}
+}
+\\
+\uncover<7->{
+&=\mathcal{K}^{k+1}(f)
+}
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Bild}
+Lineare Abbildung $f\colon V\to V$
+\[
+\operatorname{im}f
+=
+\mathcal{J}(f)
+=
+\{f(v)\;|\; v\in V\}
+\]
+\end{block}}
+\uncover<4->{%
+\begin{block}{Bild von $A^k$}
+\[
+\mathcal{J}^k(f) = \operatorname{im}f^k
+\]
+\begin{align*}
+\uncover<8->{
+\mathcal{J}^k(f)
+&=
+\operatorname{im}f^k
+=
+\operatorname{im}(f^{k}\circ f)
+}
+\\
+\uncover<9->{
+&=
+\{f^{k-1} w\;|\; w = f(v)\}
+}
+\\
+\uncover<10->{
+&\subset
+\{f^{k-1} w\;|\; w \in V\}
+}
+\\
+\uncover<11->{
+&=\mathcal{J}^{k-1}(f)
+}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/kernbilder.tex b/vorlesungen/slides/5/kernbilder.tex
new file mode 100644
index 0000000..08581ff
--- /dev/null
+++ b/vorlesungen/slides/5/kernbilder.tex
@@ -0,0 +1,68 @@
+%
+% kernbilder.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{grueneins}{rgb}{0.0,0.4,0.0}
+\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8}
+\definecolor{orangeeins}{rgb}{1.0,0.6,0.0}
+\definecolor{orangezwei}{rgb}{0.8,0.0,0.4}
+
+\begin{frame}[t]
+\frametitle{Kerne und Bilder}
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\uncover<2->{
+\begin{scope}[xshift=-4cm,yshift=1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}};
+\node[color=orangeeins] at (1.6,1.3) [right] {$\mathcal{J}^1(A)$};
+\end{scope}
+}
+
+\uncover<3->{
+\begin{scope}[xshift=-4cm,yshift=-1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}};
+\node[color=orangezwei] at (0.9,0.5) {$\mathcal{J}^2(A)$};
+\end{scope}
+}
+
+\begin{scope}[xshift=0cm,yshift=0cm]
+\uncover<1>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/leer.jpg}};
+}
+\uncover<2>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild1.jpg}};
+}
+\uncover<3>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/bild2.jpg}};
+}
+\uncover<4>{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/drei.jpg}};
+}
+\uncover<5->{
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kombiniert.jpg}};
+}
+\end{scope}
+
+\uncover<4->{
+\begin{scope}[xshift=4cm,yshift=1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern1.jpg}};
+\node[color=grueneins] at (1.0,1.3) [right] {$\mathcal{K}^1(A)$};
+\end{scope}
+}
+
+\uncover<5->{
+\begin{scope}[xshift=4cm,yshift=-1.9cm]
+\node at (0,0) {\includegraphics[width=3.6cm]{../slides/5/beispiele/kern2.jpg}};
+\node[color=gruenzwei] at (0.7,-0.6) {$\mathcal{K}^2(A)$};
+\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/kernbildintro.tex b/vorlesungen/slides/5/kernbildintro.tex
new file mode 100644
index 0000000..9fd7849
--- /dev/null
+++ b/vorlesungen/slides/5/kernbildintro.tex
@@ -0,0 +1,89 @@
+%
+% kernbildintro.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+
+\definecolor{grueneins}{rgb}{0.0,0.4,0.0}
+\definecolor{gruenzwei}{rgb}{0.0,0.4,0.8}
+\definecolor{orangeeins}{rgb}{1.0,0.6,0.0}
+\definecolor{orangezwei}{rgb}{0.8,0.0,0.4}
+
+\begin{frame}[t]
+\frametitle{Bilder und Kerne}
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\begin{scope}[xshift=-3.4cm]
+
+\only<1>{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/leer.jpg}};
+}
+\only<2-3>{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild1.jpg}};
+}
+\uncover<4->{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/bild2.jpg}};
+}
+\uncover<2->{
+	\fill[color=white,opacity=0.7]  (0.1,2.18) rectangle (4,2.64);
+	\node[color=orangeeins] at (0,2.4) [right]
+		{$\operatorname{im} A = \{Av\;|v\in\mathbb{R}^n\}$};
+}
+\uncover<4->{
+	\node[color=orangezwei] at (4,0.7) [left]
+		{$\operatorname{im} A^2 = \{A^2v\;|v\in\mathbb{R}^n\}$};
+}
+\end{scope}
+
+\begin{scope}[xshift=3.4cm]
+
+\uncover<2->{
+\fill[color=orangeeins!40] (-1,0.5) rectangle (1.8,2);
+}
+\uncover<4->{
+\fill[color=orangezwei!40] (-1.1,-1.7) rectangle (-0.,-0.3);
+}
+
+\node at (0,0) {\begin{minipage}{6cm}
+\begin{align*}
+A&={\scriptstyle\begin{pmatrix*}[r]
+  -0.979& -0.142&  0.917\\
+  -0.260& -0.643&  1.069\\
+  -0.285& -0.449&  0.823
+\end{pmatrix*}}
+\\
+\operatorname{Rang}A&=2
+\\
+\uncover<3->{
+A^2&={\scriptstyle\begin{pmatrix*}[r]
+   0.734& -0.181& -0.295\\
+   0.118& -0.029& -0.047\\
+   0.161& -0.039& -0.065
+\end{pmatrix*}}}\\
+\uncover<3->{
+\operatorname{Rang}A^2&=1}
+\end{align*}
+\end{minipage}};
+
+\only<5>{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern1.jpg}};
+}
+
+\uncover<6->{
+\node at (0,0) {\includegraphics[width=6.6cm]{../slides/5/beispiele/kern2.jpg}};
+\node[color=gruenzwei] at (-1.35,-3.0) [right] {$\ker A^2 = \{v\;|\; A^2v=0\}$};
+}
+
+\uncover<5->{
+\node[color=grueneins] at (-0.9,3.1) [right] {$\ker A = \{v\;|\; Av=0\}$};
+}
+
+\end{scope}
+
+\end{tikzpicture}
+\end{center}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/ketten.tex b/vorlesungen/slides/5/ketten.tex
new file mode 100644
index 0000000..1116a83
--- /dev/null
+++ b/vorlesungen/slides/5/ketten.tex
@@ -0,0 +1,79 @@
+%
+% ketten.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Ketten von Unterräumen}
+\begin{block}{Schachtelung}
+Die Unterräume $\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ sind geschachtelt:
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcrcrcccc}
+0       &=&\mathcal{K}^0(f)
+	&\subset&\mathcal{K}^1(f)
+	&\subset&\dots
+	&\subset&\mathcal{K}^k(f)
+	&\subset&\mathcal{K}^{k+1}(f)
+	&\subset&\dots
+	&\subset&\displaystyle\bigcup_{k=0}^\infty \mathcal{K}^k(f)
+	&=:&\mathcal{K}(f)
+\\[14pt]
+\Bbbk^n &=&\mathcal{J}^0(f)
+	&\supset&\mathcal{J}^1(f)
+	&\supset&\dots
+	&\supset&\mathcal{J}^{k}(f)
+	&\supset&\mathcal{J}^{k+1}(f)
+	&\supset&\dots
+	&\supset&\displaystyle\bigcap_{k=0}^\infty \mathcal{J}^k(f)
+	&=:&\mathcal{J}(f)
+\end{array}
+\]
+\end{block}
+\vspace{-20pt}
+\uncover<2->{%
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Abildung der Kerne}
+\vspace{-10pt}
+\begin{align*}
+f \mathcal{K}^k(f)
+&=
+\{f(v)\;|\; f^k(v) = 0\}
+\\
+&\subset
+\{ v\;|\; f^{k+1}(v)=0\}
+\\
+&=
+\mathcal{K}^{k+1}(f)
+\\
+\Rightarrow
+f\mathcal{K}(f)&= f\mathcal{K}(f)
+\quad\text{invariant}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Abbildung der Bild}
+\vspace{-10pt}
+\begin{align*}
+f\mathcal{J}^k(f)
+&=
+\{f(f^{k}(v))\;|\; v\in V\}
+\\
+&=
+\{f^{k+1}(v)\;|\; v\in V\}
+\\
+&=
+\mathcal{J}^{k+1}(f)
+\\
+\Rightarrow
+f\mathcal{J}(f)&= \mathcal{J}(f)
+\quad\text{invariant}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}}
+\end{frame}
diff --git a/vorlesungen/slides/5/konvergenzradius.tex b/vorlesungen/slides/5/konvergenzradius.tex
new file mode 100644
index 0000000..a0b4b3a
--- /dev/null
+++ b/vorlesungen/slides/5/konvergenzradius.tex
@@ -0,0 +1,109 @@
+%
+% konvergenzradius.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\setbeamercolor{column}{bg=blue!20}
+\def\punkt#1{
+	\fill[color=blue!30] #1 circle[radius=0.05];
+	\draw[color=blue] #1 circle[radius=0.05];
+}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Konvergenzradius}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Potenzreihen}
+$f\colon\mathbb{C}\to\mathbb{C}$ (komplex differenzierbar)
+\begin{equation}
+f(z) = \sum_{k=0}^\infty a_kz^k
+\label{reihe}
+\end{equation}
+\end{block}
+\vspace{-8pt}
+\uncover<2->{%
+\begin{block}{Konvergenz}
+\eqref{reihe} konvergiert für $|z| < {\color{darkgreen}R}$,
+\[
+\frac{1}{{\color{darkgreen}R}}
+=
+\limsup_{k\to\infty} |a_k|^{\frac1k}
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Polstellen}
+{\color{darkgreen}$R$} ist der Radius des grössten Kreises um $O$,
+auf dessen Rand eine
+{\color{blue}Polstelle} der Funktion $f(z)$ liegt
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\r{2.5}
+\uncover<2->{
+	\fill[color=red!20] (0,0) circle[radius=\r];
+	\draw[color=red] (0,0) circle[radius=\r];
+}
+\draw[->] (-2.6,0) -- (2.9,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-2.6) -- (0,2.9) coordinate[label={$\operatorname{Im}z$}];
+
+\uncover<2->{
+	\draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (100:\r);
+	\draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (220:\r);
+	\node[color=darkgreen] at ($0.5*(100:\r)$) [left] {$R$};
+	\node[color=darkgreen] at ($0.5*(220:\r)+(-0.1,0.1)$)
+		[below right] {$R$};
+
+	\fill[color=white] (0,0) circle[radius=0.05];
+	\draw (0,0) circle[radius=0.05];
+}
+
+\node at (2.8,2.8) {$\mathbb{C}$};
+
+\uncover<3->{
+	\punkt{(100:\r)}
+	\punkt{(220:\r)}
+
+	\begin{scope}
+		\clip (-2.6,-2.6) rectangle (2.9,2.9);
+
+		\punkt{(144.2527:2.7232)}
+		%\punkt{(226.1822:2.5164)}
+		\punkt{(173.7501:3.4140)}
+		\punkt{(267.4103,2.7668)}
+		\punkt{(137.7328:3.1683)}
+		%\punkt{(30.1155:3.3629)}
+		%\punkt{(139.1036:2.5366)}
+		\punkt{(167.4964:3.0503)}
+		\punkt{(289.2650:3.4324)}
+		\punkt{(120.1911:3.2966)}
+		%\punkt{(292.3422:2.7550)}
+		\punkt{(141.4877:2.6494)}
+		\punkt{(70.8326:2.9005)}
+		\punkt{(56.0758:3.2098)}
+		\punkt{(99.0585:3.2340)}
+		\punkt{(299.7242:2.5990)}
+		\punkt{(158.8802:2.6539)}
+		\punkt{(235.2721:2.9476)}
+		\punkt{(108.0584:2.8344)}
+		\punkt{(220.0117:2.7679)}
+
+	\end{scope}
+
+	\begin{scope}[yshift=-3.2cm,xshift=-1.0cm]
+		\punkt{(0,-0.05)}
+		\node at (0,0) [right] {$=$ Polstelle};
+	\end{scope}
+}
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/krbeispiele.tex b/vorlesungen/slides/5/krbeispiele.tex
new file mode 100644
index 0000000..b51df78
--- /dev/null
+++ b/vorlesungen/slides/5/krbeispiele.tex
@@ -0,0 +1,99 @@
+%
+% krbeispiele.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Konvergenzradius --- Beispiele}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Exponentialreihe}
+\vspace{-20pt}
+\begin{align*}
+e^z &= \sum_{k=0}^\infty \frac{z^k}{k!}
+\\
+\uncover<2->{
+\frac1k\log k!
+}
+&\uncover<3->{=\frac1k\sum_{x=1}^k {\color{blue}\log x}}
+\uncover<6->{>\frac1k\int_1^k{\color{red}\log x}\,dx}
+\\
+&
+\ifthenelse{\boolean{presentation}}{
+\only<7>{=\frac1k[x\log x -x]_1^k}
+}{}
+\only<8->{=
+\log k -1 +\frac1k}
+\uncover<9->{\to \infty\phantom{\frac1k}}
+\\
+\uncover<10->{(k!)^{\frac1k}
+&\to\infty}\uncover<11->{ \quad\Rightarrow\quad R = \infty}
+\end{align*}
+\vspace{-40pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.7]
+\uncover<4->{
+\foreach \x in {2,...,9}{
+	\fill[color=blue!20] ({\x-1},0) rectangle ({\x},{ln(\x)});
+	\draw[color=blue] ({\x-1},0) rectangle ({\x},{ln(\x)});
+	\node at ({\x-0.5},{ln(\x)}) [above] {\tiny $\log\x$};
+	\draw (\x,-0.1) -- (\x,0.1);
+	\node at (\x,0) [below] {\tiny$\x$};
+}
+\draw (1,-0.1) -- (1,0.1);
+\uncover<5->{
+\begin{scope}
+	\clip (0,-1) rectangle (9.5,2.5);
+	\fill[color=red!40,opacity=0.5] (0,0) -- (0,-1)
+		-- plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)})
+		-- (9.1,0) -- cycle;
+	\draw[color=red] plot[domain=0.1:9.1,samples=100] ({\x},{ln(\x)});
+\end{scope}
+}
+\draw[->] (-0.2,0) -- (9.4,0) coordinate[label={$x$}];
+\draw[->] (0,-1) -- (0,2.5) coordinate[label={right:$y$}];
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<12->{%
+\begin{block}{Geometrische Reihe}
+\vspace{-15pt}
+\begin{align*}
+\uncover<13->{
+\frac{1}{{\color{blue}1}-z}
+&=
+\sum_{k=0}^\infty
+z^k}
+\\
+\uncover<14->{
+a_k&=1}
+\uncover<15->{\quad\Rightarrow\quad
+|a_k|^{\frac1k}=1}
+\\
+\uncover<16->{
+\limsup_{k\to\infty} &= |a_k|^{\frac1k}=1}\uncover<17->{ = \frac1R}
+\uncover<18->{\quad\Rightarrow\quad R=1}
+\end{align*}
+%\uncover<19->{Polstelle bei $z=1$ limitiert Konvergenzradius}
+\vspace{-20pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\begin{scope}
+\clip (-2.2,-1.5) rectangle (2.2,1.5);
+\fill[color=red!20] (0,0) circle[radius=2];
+\draw[color=red] (0,0) circle[radius=2];
+\end{scope}
+\draw[->] (-2.2,0) -- (2.5,0) coordinate[label={$\operatorname{Re}z$}];
+\draw[->] (0,-1.6) -- (0,1.8) coordinate[label={right:$\operatorname{Im}z$}];
+\fill[color=blue!20] (2,0) circle[radius=0.08];
+\draw[color=blue] (2,0) circle[radius=0.08];
+\end{tikzpicture}
+\end{center}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/logarithmusreihe.tex b/vorlesungen/slides/5/logarithmusreihe.tex
new file mode 100644
index 0000000..85ba0ef
--- /dev/null
+++ b/vorlesungen/slides/5/logarithmusreihe.tex
@@ -0,0 +1,53 @@
+%
+% logarithmus.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Logarithmusreihe}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Integralgleichung}
+\vspace{-5pt}
+\begin{align*}
+\log(1+x)&=\int_0^x \frac{1}{1+t}\,dt
+\\
+&\uncover<5->{=
+\int_0^x
+1-t+t^2-t^3+\dots\,dt
+}
+\\
+\uncover<6->{
+&=
+x-\frac{x^2}2+\frac{x^3}{3}-\frac{x^4}{4}+\dots
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Geometrische Reihe}
+\vspace{-5pt}
+\begin{align*}
+\frac{1}{1-q}&=1+q+q^2+q^3+\dots
+\\
+\uncover<3->{
+\frac{1}{1+q}&=1-q+q^2-q^3+\dots
+}
+\end{align*}
+\uncover<4->{Konvergenzradius $1$}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<7->{%
+\begin{block}{Matrix-Logarithmus}
+Für $\operatorname{Sp}(A)\subset \{z\in\mathbb{C}\;|\;|z-1|<1\}$ konvergiert
+\[
+\log A
+=
+(A-I) - \frac12(A-I)^2 + \frac13(A-I)^3 - \frac14(A-I)^4 + \dots
+\]
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/motivation.tex b/vorlesungen/slides/5/motivation.tex
new file mode 100644
index 0000000..b0a1d82
--- /dev/null
+++ b/vorlesungen/slides/5/motivation.tex
@@ -0,0 +1,67 @@
+%
+% movitation.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Motivation}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Matrix $A$ analysieren}
+Matrix $A$ mit Minimalpolynom $m_A(X)$ vom
+Grad $s$
+\end{block}
+\uncover<2->{%
+\begin{block}{Faktorisieren}
+Minimalpolynom faktorisieren
+\[
+m_A(X)
+=
+(X-\mu_1)(X-\mu_2)\dots(X-\mu_s)
+\]
+\end{block}}
+\uncover<3->{%
+\begin{block}{Vertauschen}
+$\sigma\in S_s$ eine Permutation von $1,\dots,s$
+ist
+\begin{align*}
+m_A(X)
+&=
+(X-\mu_{\sigma(1)})
+%(X-\mu_{\sigma(2)})
+\dots
+(X-\mu_{\sigma(s)})
+\\
+0
+&=
+(A-\mu_{\sigma(1)})
+%(A-\mu_{\sigma(2)})
+\dots
+(A-\mu_{\sigma(s)})
+\end{align*}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<4->{%
+\begin{block}{Bedingung für $\mu_k$}
+Permutation wählen so dass $\mu_k$ an erster Stelle steht:
+\[
+0=(A-\mu_k) \prod_{i\ne k}(A-\mu_i) v
+\]
+für alle $v\in\Bbbk^n$.
+\end{block}}
+\uncover<5->{%
+\begin{block}{Eigenwerte}
+Nur diejenigen ${\color{red}\mu}$ sind möglich, für die es $v\in\Bbbk^n$
+gibt mit
+\[
+(A-\mu)v = 0
+\Rightarrow Av = {\color{red}\mu} v
+\]
+Eigenwertbedingung
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/nilpotent.tex b/vorlesungen/slides/5/nilpotent.tex
new file mode 100644
index 0000000..ca38c40
--- /dev/null
+++ b/vorlesungen/slides/5/nilpotent.tex
@@ -0,0 +1,190 @@
+%
+% nilpotent.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\feld#1{
+	\fill[color=red!20] (#1,0) rectangle ({#1+1},12);
+}
+\begin{frame}[t]
+\frametitle{$\mathcal{J}^k(f)$ und $\mathcal{K}^k(f)$ für nilpotente Matrizen}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.42\textwidth}
+Matrix mit dem dargestellten Verlauf von
+${\color{red}\dim\mathcal{K}^k(A)}$
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.42]
+
+\only<2->{
+	\feld{0}
+	\feld{1}
+	\feld{2}
+	\feld{3}
+}
+\only<2->{ \feld{4} }
+\only<2->{ \feld{6} }
+\ifthenelse{\boolean{presentation}}{
+\only<3->{ \feld{5} }
+\only<3->{ \feld{7} }
+\only<4->{ \feld{8} }
+\only<5->{ \feld{9} }
+\only<6->{ \feld{10} }
+\only<7->{ \feld{11} }
+
+\only<1>{ \node at (6,0) [below] {$k=0$}; }
+}{}
+\only<2>{ \node at (6,0) [below] {$k=1$}; }
+\ifthenelse{\boolean{presentation}}{
+\only<3>{ \node at (6,0) [below] {$k=2$}; }
+\only<4>{ \node at (6,0) [below] {$k=3$}; }
+\only<5>{ \node at (6,0) [below] {$k=4$}; }
+\only<6>{ \node at (6,0) [below] {$k=5$}; }
+\only<7>{ \node at (6,0) [below] {$k=6$}; }
+}{}
+
+\draw (0,0) rectangle (12,12);
+\ifthenelse{\boolean{presentation}}{
+\only<1>{
+	\foreach \x in {1,...,12}{
+		\node at ({\x-0.5},{12-\x+0.5}) {$1$};
+	}
+}
+}{}
+\only<2->{
+	\foreach \x in {1,...,12}{
+		\node at ({\x-0.5},{12-\x+0.5}) {$0$};
+	}
+}
+\only<2>{
+	\foreach \x in {7,...,11}{
+		\node at ({\x+0.5},{12-\x+0.5}) {$1$};
+	}
+}
+\ifthenelse{\boolean{presentation}}{
+\only<3->{
+	\foreach \x in {7,...,11}{
+		\node at ({\x+0.5},{12-\x+0.5}) {$0$};
+	}
+}
+\only<3>{
+	\foreach \x in {8,...,11}{
+		\node at ({\x+0.5},{13-\x+0.5}) {$1$};
+	}
+}
+\only<4->{
+	\foreach \x in {8,...,11}{
+		\node at ({\x+0.5},{13-\x+0.5}) {$0$};
+	}
+}
+\only<4>{
+	\foreach \x in {9,...,11}{
+		\node at ({\x+0.5},{14-\x+0.5}) {$1$};
+	}
+}
+\only<5->{
+	\foreach \x in {9,...,11}{
+		\node at ({\x+0.5},{14-\x+0.5}) {$0$};
+	}
+}
+\only<5>{
+	\foreach \x in {10,...,11}{
+		\node at ({\x+0.5},{15-\x+0.5}) {$1$};
+	}
+}
+\only<6->{
+	\foreach \x in {10,...,11}{
+		\node at ({\x+0.5},{15-\x+0.5}) {$0$};
+	}
+}
+\only<6>{
+	\foreach \x in {11,...,11}{
+		\node at ({\x+0.5},{16-\x+0.5}) {$1$};
+	}
+}
+\only<7->{
+	\foreach \x in {11,...,11}{
+		\node at ({\x+0.5},{16-\x+0.5}) {$0$};
+	}
+}
+}{}
+\draw[line width=0.1pt]
+	(0,11) -- (2,11) -- (2,9) -- (4,9) -- (4,6) -- (12,6);
+\draw[line width=0.1pt]
+	(1,12) -- (1,10) -- (3,10) -- (3,8) -- (6,8) -- (6,0);
+\only<2>{
+	\node at (5.5,7.5) {$1$};
+}
+\ifthenelse{\boolean{presentation}}{
+\only<3->{
+	\node at (5.5,7.5) {$0$};
+}
+}{}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.56\textwidth}
+\vspace{-15pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\def\pfad{
+	(0,0) -- (1,3) -- (2,4) -- (3,4.5) -- (4,5) -- (5,5.5) -- (6,6)
+}
+\fill[color=orange!20] \pfad -- (0,6) -- cycle;
+\fill[color=darkgreen!20] \pfad -- (6,0) -- cycle;
+\foreach \y in {0.5,1,...,5.75}{
+	\draw[line width=0.1pt] (0,\y) -- (6,\y);
+}
+\draw[line width=1.4pt] \pfad;
+\draw[->] (-0.1,6) -- (6.5,6); \node at (-0.1,6) [left] {$n$};
+\draw[->] (-0.1,0) -- (6.5,0); \node at (-0.1,0) [left] {$0$};
+\fill (0,0) circle[radius=0.05];
+\fill (1,3) circle[radius=0.05];
+\fill (2,4) circle[radius=0.05];
+\fill (3,4.5) circle[radius=0.05];
+\fill (4,5) circle[radius=0.05];
+\fill (5,5.5) circle[radius=0.05];
+\fill (6,6) circle[radius=0.05];
+\ifthenelse{\boolean{presentation}}{
+\only<1>{
+	\fill[color=red] (0,0) circle[radius=0.08];
+}
+}{}
+\only<2>{
+	\fill[color=red] (1,3) circle[radius=0.08];
+	\draw[color=red] (0,3) -- (1,3);
+	\node[color=red] at (0,3) [left] {$6$};
+}
+\ifthenelse{\boolean{presentation}}{
+\only<3>{
+	\fill[color=red] (2,4) circle[radius=0.08];
+	\draw[color=red] (0,4) -- (2,4);
+	\node[color=red] at (0,4) [left] {$8$};
+}
+\only<4>{
+	\fill[color=red] (3,4.5) circle[radius=0.08];
+	\draw[color=red] (0,4.5) -- (3,4.5);
+	\node[color=red] at (0,4.5) [left] {$9$};
+}
+\only<5>{
+	\fill[color=red] (4,5.0) circle[radius=0.08];
+	\draw[color=red] (0,5.0) -- (4,5.0);
+	\node[color=red] at (0,5.0) [left] {$10$};
+}
+\only<6>{
+	\fill[color=red] (5,5.5) circle[radius=0.08];
+	\draw[color=red] (0,5.5) -- (5,5.5);
+	\node[color=red] at (0,5.5) [left] {$11$};
+}
+\only<7>{
+	\fill[color=red] (6,6.0) circle[radius=0.08];
+}
+}{}
+\draw[color=white] (-0.7,0) -- (-0.7,6);
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/normal.tex b/vorlesungen/slides/5/normal.tex
new file mode 100644
index 0000000..7245608
--- /dev/null
+++ b/vorlesungen/slides/5/normal.tex
@@ -0,0 +1,69 @@
+%
+% normal.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Normale Operatoren}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Frage}
+$f,g\colon \mathbb{C}\to\mathbb{C}$.
+\\
+In welchen Punkten müssen $f$ und $g$ übereinstimmen, damit
+$f(A)=g(A)$?
+\end{block}
+\uncover<2->{%
+\begin{block}{Definition $f(A)$}
+$f$ durch eine Folge von Polynomen
+appoximieren: $p_n\to f$
+\[
+f(A) = \lim_{n\to\infty}p_n(A)
+\]
+\end{block}}
+\vspace{-15pt}
+\uncover<3->{%
+\begin{block}{Vermutung}
+Falls $f(z)=g(z)$ für $z\in\operatorname{Sp}(A)$,
+dann ist $f(A)=g(A)$
+
+\smallskip
+\uncover<4->{%
+{\usebeamercolor[fg]{title}Stimmt für: } $A$ diagonalisierbar
+}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<5->{%
+\begin{block}{Beispiel}
+\[
+A=\begin{pmatrix}2&1\\0&2\end{pmatrix}, \quad
+\operatorname{Sp}(A)=\{2\}
+\]
+\uncover<6->{%
+\begin{align*}
+f(z)&=(z-2)^2 &g(z)&=z-2
+\\
+\uncover<7->{
+f(A)&=0&g(A)&=\begin{pmatrix}0&1\\0&0\end{pmatrix}
+}
+\end{align*}}
+\end{block}}
+\vspace{-18pt}
+\uncover<8->{%
+\begin{block}{Normal}
+$A$ heisst {\em normal}, wenn $AA^*=A^*A$
+\begin{itemize}
+\item<9->
+symmetrische Matrizen: $A=A^*$
+\item<10->
+unitäre Matrizen: $A^*=A^{-1}\Rightarrow
+AA^*=AA^{-1}=A^{-1}A=A^*A$
+\end{itemize}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/normalnilp.tex b/vorlesungen/slides/5/normalnilp.tex
new file mode 100644
index 0000000..9457136
--- /dev/null
+++ b/vorlesungen/slides/5/normalnilp.tex
@@ -0,0 +1,237 @@
+%
+% normalnilp.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\sx{1.9}
+\def\sy{0.6}
+\def\punkt#1#2#3{
+	\foreach \y in {0,...,#2}{
+	}
+}
+\def\block#1#2{
+	\fill[rounded corners=2pt,color=white]
+		({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05});
+	\draw[rounded corners=2pt]
+		({-#1*\sx-0.4},-0.05) rectangle ({-#1*\sx+0.4},{#2*\sy+0.05});
+}
+\def\teilmenge#1#2#3{
+	\fill[rounded corners=2pt,color=white]
+		({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=gray]
+		({-#1*\sx-0.35},{#2*\sy}) rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\rot#1#2#3{
+	\fill[rounded corners=2pt,color=red!20]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=red]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\hellblau#1#2#3{
+	\fill[rounded corners=2pt,color=blue!20]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+	\draw[rounded corners=2pt,color=blue!40]
+		({-#1*\sx-0.35},{#2*\sy+0.00})
+			rectangle ({-#1*\sx+0.35},{#3*\sy+0.00});
+}
+\def\punkt#1#2{
+	\fill[color=blue] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08];
+}
+\def\bildpunkt#1#2{
+	\fill[color=blue!40] ({-#1*\sx},{(#2-0.5)*\sy}) circle[radius=0.08];
+}
+\def\pfeil#1#2#3{
+	\draw[->,color=blue,shorten >= 0.1cm,shorten <= 0.1cm] 
+	({-#1*\sx},{(#2-0.5)*\sy}) 
+	--
+	({-(#1-1)*\sx},{(#3-0.5)*\sy}) ;
+}
+\begin{frame}[t]
+\frametitle{Normalform einer nilpotenten Matrix}
+{\usebeamercolor[fg]{title}$A^l=0$ $\Rightarrow$ finde eine ``gute'' Basis}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\vspace{-25pt}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+
+\fill[color=darkgreen!20,rounded corners=2pt]
+	({-3*\sx+0.35},0) -- (-0.35,0)
+	--
+	({-1*\sx+0.35},{4*\sy}) -- ({-1*\sx-0.35},{4*\sy})
+	--
+	({-2*\sx+0.35},{7*\sy}) -- ({-2*\sx-0.35},{7*\sy})
+	--
+	({-3*\sx+0.35},{8*\sy}) -- cycle;
+
+\block{0}{0}
+
+\block{1}{4}
+\uncover<10->{
+	\rot{1}{0}{1}
+	\node[color=red] at ({-1*\sx-0.28},{0.5*\sy}) [left] {$\mathcal{C}_{l-2}$};
+}
+\uncover<8->{
+	\hellblau{1}{1}{3}
+}
+\uncover<4->{
+	\hellblau{1}{3}{4}
+}
+
+\block{2}{7}
+\uncover<4->{
+	\hellblau{2}{6}{7}
+}
+\uncover<6->{
+	\rot{2}{4}{6}
+	\node[color=red] at ({-2*\sx-0.28},{5*\sy}) [left] {$\mathcal{C}_{l-1}$};
+}
+\teilmenge{2}{0}{4}
+
+\block{3}{8}
+\uncover<2->{
+	\rot{3}{7}{8}
+	\node[color=red] at ({-3*\sx-0.28},{7.5*\sy}) [left] {$\mathcal{C}_l$};
+}
+\teilmenge{3}{0}{7}
+
+\uncover<3->{
+	\punkt{3}{8}
+}
+\uncover<4->{
+	\pfeil{3}{8}{7}
+	\bildpunkt{2}{7}
+	\pfeil{2}{7}{4}
+	\bildpunkt{1}{4}
+}
+
+\uncover<7->{
+	\punkt{2}{5}
+	\punkt{2}{6}
+}
+\uncover<8->{
+	\pfeil{2}{5}{2}
+	\bildpunkt{1}{3}
+	\pfeil{2}{6}{3}
+	\bildpunkt{1}{2}
+}
+
+\uncover<11->{
+\punkt{1}{1}
+}
+
+\node at ({-3*\sx},0) [below] {$\mathcal{K}^l(A)\mathstrut$};
+\node at ({-2*\sx},0) [below] {$\mathcal{K}^{l-1}(A)\mathstrut$};
+\node at ({-1.45*\sx},0) [below] {$\dots\mathstrut$};
+\node at ({-1*\sx},0) [below] {$\mathcal{K}^1(A)\mathstrut$};
+\node at ({-0*\sx},0) [below] {$0=\mathcal{K}^0(A)\mathstrut$};
+\node[color=gray] at ({-2*\sx},{2*\sy}) [rotate=90] {$\mathcal{K}^1(A)$};
+\node[color=gray] at ({-3*\sx},{3.5*\sy}) [rotate=90] {$\mathcal{K}^{l-1}(A)$};
+\foreach \x in {0,1,2}{
+	\draw[->,shorten >= 0.1cm, shorten <= 0.1cm]
+		({-(\x+1)*\sx},{8.7*\sy}) -- ({-(\x)*\sx},{8.7*\sy});
+	\node at ({-(\x+0.5)*\sx},{8.7*\sy}) [above] {$A$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\vspace{-30pt}
+\begin{enumerate}
+\item<2->	\(
+	\mathcal{K}^l(A)=\mathcal{K}^{l-1}\oplus {\color{red}\mathcal{C}_l}
+	\)
+\item<3->	\(
+	{\color{blue}b_1}\in{\color{red}\mathcal{C}_l}
+	\)
+\item<4->	\(
+	\mathcal{B}_l
+	=
+	\{{\color{blue}b_1},{\color{blue!40}Ab_1},{\color{blue!40}A^2b_1},\dots,
+	{\color{blue!40}A^{l-1}b_1}\}
+	\)
+\item<5->	\(
+	\mathcal{K}^{l-1}(A)
+	=
+	\mathcal{K}^{l-2}(A)
+	\oplus
+	{\color{red}\mathcal{C}_{l-1}}
+	\oplus
+	{\color{blue}A\mathcal{C}_l}
+	\)
+\item<6->	\(
+	{\color{blue}b_2},{\color{blue}b_3}\in{\color{red}\mathcal{C}_{l-1}}
+	\)
+\item<7->	\(
+	\mathcal{B}_{l-1}
+	=
+	\{
+	{\color{blue}b_2},{\color{blue}b_3},
+	{\color{blue!40}Ab_2}, {\color{blue!40}Ab_3},\dots
+	\}
+	\)
+\item<8-> \dots
+\end{enumerate}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.4]
+
+\uncover<2-4>{
+	\fill[color=red!20] (2,0) rectangle (3,8);
+}
+\uncover<4->{
+	\fill[color=blue!20] (0,6) rectangle (2,8);
+}
+\uncover<5->{
+	\fill[color=red!20] (2,5) rectangle (3,8);
+	\node[color=blue] at (2.5,6.5) {$1$};
+	\node[color=blue] at (1.5,7.5) {$1$};
+	\node[color=gray] at (0.5,7.5) {$0$};
+	\node[color=gray] at (1.5,6.5) {$0$};
+	\node[color=gray] at (2.5,5.5) {$0$};
+	\draw[color=gray] (0.05,5.05) rectangle (2.95,7.95);
+}
+
+\uncover<6-8>{
+	\fill[color=red!20] (4,0) rectangle (5,8);
+	\fill[color=red!20] (6,0) rectangle (7,8);
+}
+\uncover<8->{
+	\fill[color=blue!20] (3,4) rectangle (4,5);
+	\fill[color=blue!20] (5,2) rectangle (6,3);
+}
+\uncover<9->{
+	\fill[color=red!20] (4,3) rectangle (5,5);
+	\node[color=blue] at (4.5,4.5) {$1$};
+	\node[color=gray] at (3.5,4.5) {$0$};
+	\node[color=gray] at (4.5,3.5) {$0$};
+	\draw[color=gray] (3.05,3.05) rectangle (4.95,4.95);
+	\fill[color=red!20] (6,1) rectangle (7,3);
+	\node[color=blue] at (6.5,2.5) {$1$};
+	\node[color=gray] at (5.5,2.5) {$0$};
+	\node[color=gray] at (6.5,1.5) {$0$};
+	\draw[color=gray] (5.05,1.05) rectangle (6.95,2.95);
+}
+
+\uncover<10>{
+	\fill[color=red!20] (7,0) rectangle (8,8);
+}
+\uncover<11->{
+	\fill[color=red!20] (7,0) rectangle (8,1);
+	\node[color=gray] at (7.5,0.5) {$0$};
+	\draw[color=gray] (7.05,0.05) rectangle (7.95,0.95);
+}
+
+\draw (0,0) rectangle (8,8);
+\node at (-0.1,4) [left] {$A=$};
+
+\end{tikzpicture}
+\end{center}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/potenzreihenmethode.tex b/vorlesungen/slides/5/potenzreihenmethode.tex
new file mode 100644
index 0000000..0c3503d
--- /dev/null
+++ b/vorlesungen/slides/5/potenzreihenmethode.tex
@@ -0,0 +1,93 @@
+%
+% potenzreihenmethode.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Potenzreihenmethode}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Lineare Differentialgleichung}
+\vspace{-12pt}
+\begin{align*}
+y'&=ay&&\Rightarrow&y'-ay&=0
+\\
+y(0)&=C
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<2->{%
+\begin{block}{Potenzreihenansatz}
+\vspace{-12pt}
+\begin{align*}
+y(x)
+&=
+a_0+ a_1x + a_2x^2 + \dots
+\\
+y(0)&=a_0=C
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\uncover<3->{%
+\begin{block}{Lösung}
+\vspace{-12pt}
+\[
+\arraycolsep=1.4pt
+\begin{array}{rcrcrcrcrcr}
+\uncover<3->{ y'(x)}
+	\uncover<5->{
+	&=&\phantom{(} a_1\phantom{\mathstrut-aa_0)}
+	&+& 2a_2\phantom{\mathstrut-aa_1)}x
+		&+& 3a_3\phantom{\mathstrut-aa_2)}x^2
+			&+& 4a_4\phantom{\mathstrut-aa_3)}x^3
+				&+& \dots}\\
+\uncover<3->{-ay(x)}
+	\uncover<6->{
+	&=&\mathstrut-aa_0 \phantom{)}
+	&-&  aa_1\phantom{)}x
+		&-&  aa_2\phantom{)}x^2
+			&-&  aa_3\phantom{)}x^3
+				&-& \dots}\\[2pt]
+\hline
+\\[-10pt]
+\uncover<3->{0}
+	\uncover<7->{
+	&=&(a_1-aa_0)
+	&+& (2a_2-aa_1)x
+		&+& (3a_3-aa_2)x^2
+			&+& (4a_4-aa_3)x^3
+				&+& \dots}\\
+\end{array}
+\]
+\begin{align*}
+\uncover<4->{
+a_0&=C}\uncover<8->{,
+\quad
+a_1=aa_0=aC}\uncover<9->{,
+\quad
+a_2=\frac12a^2C}\uncover<10->{,
+\quad
+a_3=\frac16a^3C}\uncover<11->{,
+\dots
+a_k=\frac1{k!}a^kC}
+\hspace{3cm}
+\\
+\uncover<4->{
+\Rightarrow y(x) &= C}\uncover<8->{+Cax}\uncover<9->{ + C\frac12(ax)^2}
+\uncover<10->{ + C \frac16(ac)^3}
+\uncover<11->{ + \dots+C\frac{1}{k!}(ax)^k+\dots}
+\ifthenelse{\boolean{presentation}}{
+\only<12>{
+=
+C\sum_{k=0}^\infty \frac{(ax)^k}{k!}}
+}{}
+\uncover<13->{=
+Ce^{ax}}
+\end{align*}
+\end{block}}
+\end{frame}
diff --git a/vorlesungen/slides/5/reellenormalform.tex b/vorlesungen/slides/5/reellenormalform.tex
new file mode 100644
index 0000000..4ceabe9
--- /dev/null
+++ b/vorlesungen/slides/5/reellenormalform.tex
@@ -0,0 +1,115 @@
+%
+% reellenormalform.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Reelle Normalform}
+$A\in M_n(\mathbb{R})\subset M_n(\mathbb{C})$ hat reelle und Paare von
+konjugiert komplexen Eigenwerten
+\medskip
+
+$\Rightarrow$ Konjugiert komplexe Eigenvektoren $v$ und $\overline{v}$,
+$x=\operatorname{Re}v$ und $y=\operatorname{Im}v$
+\begin{align*}
+\only<-2>{
+\begin{pmatrix}
+Av\\
+A\overline v
+\end{pmatrix}
+=
+\begin{pmatrix}
+Ax+Ay J \\
+Ax-Ay J
+\end{pmatrix}
+&=
+\begin{pmatrix}
+\lambda v\\
+\overline{\lambda}\overline{v}
+\end{pmatrix}
+=
+\begin{pmatrix}
+a+bJ & 0 \\
+  0  & a-bJ
+\end{pmatrix}
+\begin{pmatrix}
+x+ yJ\\
+x- yJ
+\end{pmatrix}
+\\
+}
+\only<2-3>{
+\begin{pmatrix}
+Ax&-Ay\\
+Ay& Ax\\
+Ax& Ay\\
+-Ay&Ax
+\end{pmatrix}
+&=
+\begin{pmatrix}
+a&-b& 0& 0\\
+b& a& 0& 0\\
+0& 0& a& b\\
+0& 0&-b& a
+\end{pmatrix}
+\begin{pmatrix}
+x&-y\\
+y& x\\
+x& y\\
+-y&x
+\end{pmatrix}
+\\
+}
+\only<3-4>{
+\ifthenelse{\boolean{presentation}}{
+\begin{pmatrix}
+Ax&-Ay\\
+Ax& Ay\\
+Ay& Ax\\
+-Ay&Ax
+\end{pmatrix}
+&
+=
+\begin{pmatrix}
+a& 0&-b& 0\\
+0& a& 0& b\\
+b& 0& a& 0\\
+0&-b& 0& a
+\end{pmatrix}
+\begin{pmatrix}
+x&-y\\
+x& y\\
+y& x\\
+-y&x
+\end{pmatrix}
+\Rightarrow
+\\
+}{}
+}
+\only<4->{
+Ax &= ax -by \\
+Ay &= bx +ay
+}
+\end{align*}
+\uncover<5->{%
+D.h. in Basis $x=\operatorname{Re}v,y=\operatorname{Im}v$ hat $A$ die Matrix
+$\begin{pmatrix}a&-b\\b&a\end{pmatrix}$}
+\uncover<6->{%
+\[
+\text{
+Reeller
+Jordan-Block:
+}
+\qquad
+J_{\lambda,\overline{\lambda}}
+=
+\begin{pmatrix}
+a&-b&1& 0&0& 0\\
+b& a&0& 1&0& 0\\
+ &  &a&-b&1& 0\\
+ &  &b& a&0& 1\\
+ &  & &  &a&-b\\
+ &  & &  &b& a
+\end{pmatrix}
+\]}
+\end{frame}
diff --git a/vorlesungen/slides/5/satzvongelfand.tex b/vorlesungen/slides/5/satzvongelfand.tex
new file mode 100644
index 0000000..3cf8710
--- /dev/null
+++ b/vorlesungen/slides/5/satzvongelfand.tex
@@ -0,0 +1,89 @@
+%
+% satzvongelfand.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{0pt}
+\setlength{\belowdisplayskip}{0pt}
+\setbeamercolor{block body}{bg=blue!20}
+\setbeamercolor{block title}{bg=blue!20}
+\frametitle{Satz von Gelfand}
+{\usebeamercolor[fg]{title}Behauptung:} $\varrho(A)=\pi(A)$\uncover<2->{,
+$A(\varepsilon) = \displaystyle\frac{A}{\varrho(A)+\varepsilon}$}\uncover<3->{,
+$\varrho(A(\varepsilon))=\displaystyle\frac{\varrho(A)}{\varrho(A)+\varepsilon}
+\uncover<4->{=\frac{1}{1+\varepsilon/\varrho(A)}}$}
+
+\uncover<5->{%
+%{\usebeamercolor[fg]{title}Beweisidee:}
+%$\displaystyle\pi\biggl(\frac{A}{\varrho(A)+\epsilon}\biggr)
+%=
+%\frac{\pi(A)}{\varrho(A)+\epsilon}$ berechnen
+\vspace{-5pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{$\varepsilon < 0$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<6->{
+\varrho(A(\varepsilon))&>1}\uncover<7->{\quad\Rightarrow\quad \|A(\varepsilon)^k\|\to \infty}
+\\
+\uncover<8->{\|A(\varepsilon)^k\| &\ge m\varrho(A(\varepsilon))^k}
+\\
+\uncover<9->{\|A(\varepsilon)^k\|^{\frac1k} &\ge m^{\frac1k} \varrho(A(\varepsilon))}
+\\
+\uncover<10->{\pi(A) &\ge \lim_{k\to\infty}m^{\frac1k}\varrho(A(\varepsilon))}
+\\
+&\uncover<11->{= \varrho(A(\varepsilon))}\uncover<12->{ > 1}
+\\
+\uncover<13->{\frac{ \pi(A(\varepsilon))}{\varrho(A)+\varepsilon} &> 1}
+\\
+\uncover<14->{
+\pi(A) &> \varrho(A)+\varepsilon
+}
+\end{align*}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{$\varepsilon > 0$}
+\vspace{-10pt}
+\begin{align*}
+\uncover<16->{
+\varrho(A(\varepsilon)) &<1}
+\uncover<17->{\quad\Rightarrow\quad \|A(\varepsilon)^k\| \to 0}
+\\
+\uncover<18->{\|A(\varepsilon)^k\|
+&\le M\varrho(A(\varepsilon))^k}
+\\
+\uncover<19->{
+\|A(\varepsilon)^k\|^{\frac1k}
+&\le M^{\frac1k}\varrho(A(\varepsilon))
+}
+\\
+\uncover<20->{
+\pi(A(\varepsilon))
+&\le
+\varrho(A(\varepsilon)) \lim_{k\to\infty} M^{\frac1k}
+}
+\\
+&\uncover<21->{= \varrho(A(\varepsilon))}
+\uncover<22->{ < 1}
+\\
+\uncover<23->{\frac{\pi(A)}{\varrho(A)+\varepsilon}&< 1}
+\\
+\uncover<24->{\pi(A)&< \varrho(A) + \varepsilon}
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}}
+\uncover<15->{%
+\vspace{2pt}
+{\usebeamercolor[fg]{title}Folgerung:}
+$\varrho(A)-\varepsilon < \pi(A) \uncover<25->{< \varrho(A)+\varepsilon}\quad\forall\varepsilon>0
+\uncover<26->{
+\qquad\Rightarrow\qquad
+\varrho(A)=\pi(A)}$
+}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/spektralgelfand.tex b/vorlesungen/slides/5/spektralgelfand.tex
new file mode 100644
index 0000000..9342cd6
--- /dev/null
+++ b/vorlesungen/slides/5/spektralgelfand.tex
@@ -0,0 +1,190 @@
+%
+% spektralgelfand.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\def\eigenwert#1#2{
+	\fill[color=blue!30] (#1:#2) circle[radius=0.05];
+	\draw[color=blue] (#1:#2) circle[radius=0.05];
+}
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spektral- und Gelfand-Radius}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth}
+\begin{block}{Spektralradius}
+\vspace{-10pt}
+\[
+\varrho(A)
+=
+\sup\{|\lambda|\;|\; \text{{\color{blue}$\lambda$} ist EW von $A$}\}
+\]
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\uncover<5->{
+	\fill[color=red!30] (0,0) circle[radius=2.2];
+	\draw[color=red] (0,0) circle[radius=2.2];
+}
+
+\uncover<3->{
+	\eigenwert{190.46}{1.3365}
+	%\eigenwert{52.663}{2.1819}
+	\eigenwert{281.94}{1.7305}
+	\eigenwert{21.29}{1.0406}
+	\eigenwert{69.511}{1.56}
+	\eigenwert{63.365}{1.3535}
+	\eigenwert{281.43}{0.31994}
+	\eigenwert{313.1}{1.5419}
+	\eigenwert{118.14}{1.1966}
+	\eigenwert{195.75}{0.41156}
+	\eigenwert{233.42}{1.5613}
+	\eigenwert{25.203}{1.1936}
+	\eigenwert{53.375}{1.4886}
+	\eigenwert{346.13}{2.1073}
+	\eigenwert{246.47}{1.124}
+	\eigenwert{35.451}{1.99}
+	\eigenwert{212.43}{1.9708}
+	\eigenwert{58.479}{0.61602}
+	\eigenwert{344.37}{1.6107}
+	\eigenwert{305.42}{1.7075}
+	\eigenwert{29.693}{0.28791}
+	\eigenwert{195.82}{0.63079}
+	\eigenwert{209.71}{0.25669}
+	\eigenwert{51.355}{0.7247}
+	\eigenwert{356.43}{1.0867}
+	\eigenwert{33.119}{0.7328}
+	\eigenwert{73.131}{1.5021}
+	\eigenwert{345.67}{0.37564}
+	\eigenwert{76.52}{0.71763}
+	%\eigenwert{197.04}{2.1431}
+	\eigenwert{217.87}{1.7704}
+	\eigenwert{172.93}{1.1204}
+	\eigenwert{339.19}{1.5305}
+	\eigenwert{272.86}{2.04}
+	\eigenwert{168.8}{1.6289}
+	\eigenwert{248.68}{0.70879}
+	\eigenwert{237.98}{0.71097}
+	\eigenwert{81.411}{1.8461}
+	\eigenwert{224.65}{1.0827}
+	\eigenwert{357.54}{0.291}
+	\eigenwert{325.26}{1.2778}
+	\eigenwert{150.97}{0.32358}
+	\eigenwert{260.68}{1.4077}
+	\eigenwert{116.29}{1.0715}
+	\eigenwert{358.25}{0.99667}
+	\eigenwert{276.2}{0.077375}
+	\eigenwert{316.16}{0.77763}
+	\eigenwert{69.398}{1.2818}
+	\eigenwert{353.5}{0.74099}
+	\eigenwert{4.7935}{1.391}
+	\eigenwert{136.98}{1.7572}
+	\eigenwert{45.62}{1.9649}
+	\eigenwert{299.96}{0.19199}
+	\eigenwert{187.32}{0.63805}
+	\eigenwert{272.88}{1.1467}
+	\eigenwert{231.85}{1.5763}
+	\eigenwert{124.24}{0.77024}
+	\eigenwert{196.24}{2.0375}
+	\eigenwert{186.33}{1.0656}
+	%\eigenwert{22.812}{2.1616}
+	\eigenwert{37.982}{0.038956}
+	\eigenwert{142.36}{1.7944}
+	\eigenwert{56.863}{1.8952}
+	\eigenwert{4.6281}{1.1857}
+	\eigenwert{71.674}{0.07642}
+	\eigenwert{94.049}{1.8985}
+	\eigenwert{97.294}{0.23412}
+	\eigenwert{84.739}{0.31209}
+	\eigenwert{147.42}{1.8434}
+	\eigenwert{160.67}{0.76956}
+	\eigenwert{292.5}{0.85697}
+	\eigenwert{308.1}{1.7061}
+	\eigenwert{68.669}{2.111}
+	\eigenwert{86.866}{1.1271}
+	\eigenwert{124.72}{1.3019}
+	\eigenwert{267.36}{0.7462}
+	\eigenwert{295.78}{1.0425}
+	\eigenwert{44.972}{0.65363}
+	\eigenwert{34.534}{1.2817}
+	\eigenwert{357.78}{2.0592}
+	\eigenwert{147.52}{0.020535}
+	%\eigenwert{28.502}{2.1964}
+	\eigenwert{343.48}{2.0968}
+	\eigenwert{129.96}{0.80371}
+	\eigenwert{254.75}{1.5775}
+	\eigenwert{89.91}{0.88605}
+	\eigenwert{20.35}{0.66065}
+	\eigenwert{60.382}{1.7585}
+	\eigenwert{158.87}{0.68399}
+	\eigenwert{328.44}{1.504}
+	\eigenwert{189.41}{0.33879}
+	\eigenwert{273.47}{0.11109}
+	\eigenwert{285.99}{0.66704}
+	\eigenwert{311.42}{2.0266}
+	\eigenwert{32.636}{0.5713}
+	\eigenwert{221.35}{2.1329}
+	\eigenwert{50.983}{1.1957}
+	\eigenwert{53.298}{1.2982}
+	\eigenwert{101.4}{1.9051}
+	\eigenwert{71.999}{0.25671}
+}
+
+\uncover<2->{
+	\draw[->] (-2.4,0) -- (2.7,0)
+		coordinate[label={$\operatorname{Re}z$}];
+	\draw[->] (0,-2.4) -- (0,2.5)
+		coordinate[label={right:$\operatorname{Im}z$}];
+}
+\uncover<4->{
+	\fill[color=darkgreen] (0,0) circle[radius=0.05];
+	\draw[->,color=darkgreen,shorten >= 0.05cm] (0,0) -- (150:2.2);
+	\node[color=darkgreen] at ($(150:1.85)+(0.4,0)$)
+		[below left] {$\varrho(A)$};
+}
+\uncover<3->{
+	\eigenwert{150}{2.2}
+}
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<6->{%
+\begin{block}{Gelfand-Radius}
+\[
+\pi(A)
+=
+\lim_{k\to\infty} \|A^k\|^{\frac{1}{k}}
+\]
+\end{block}}
+\vspace{-8pt}
+\uncover<7->{%
+\begin{block}{Konvergenz der Neumann-Reihe}
+$
+\uncover<8->{t<1/\pi(A)\;}
+\uncover<10->{\Rightarrow\; \exists q}
+\uncover<11->{,N}$
+\begin{align*}
+\uncover<9->{ t\pi(A) & \only<10->{< q} < 1 }
+\\
+\uncover<11->{ \|(tA)^k\|^{\frac1k} &\le q }
+\\
+\uncover<12->{
+\|(tA)^k\|
+&\le
+(t\pi(A))^k<q^k
+}
+\end{align*}
+\uncover<11->{für $k>N$.}
+\uncover<13->{
+$\Rightarrow$
+$(1-tA)^{-1}=\displaystyle\sum_{k=0}^\infty (tA)^k$ konvergiert für $t<1/\pi(A)$
+}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/spektrum.tex b/vorlesungen/slides/5/spektrum.tex
new file mode 100644
index 0000000..6cbdd7f
--- /dev/null
+++ b/vorlesungen/slides/5/spektrum.tex
@@ -0,0 +1,76 @@
+%
+% spektrum.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Spektrum}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+$A\colon V\to V$ beschränkter Operator zwischen Banach-Räumen
+\[
+\operatorname{Sp}A
+=
+\left\{
+\lambda\in\mathbb{C}
+\;\left|\;
+\begin{minipage}{2cm}\raggedright
+$A-\lambda I$ nicht invertierbar
+\end{minipage}
+\right.
+\right\}
+\]
+\end{block}
+\uncover<2->{%
+\begin{block}{Endlichdimensionale Räume}
+\vspace{-15pt}
+\begin{align*}
+&\lambda\in\operatorname{Sp}A
+\\
+\uncover<3->{
+\Leftrightarrow\quad&\text{$(A-\lambda I)$ nicht invertierbar}
+}
+\\
+\uncover<4->{
+\Leftrightarrow\quad&\text{$(A-\lambda I)$ singulär}
+}
+\\
+\uncover<5->{
+\Leftrightarrow\quad&\ker(A-\lambda I)\ne 0
+}
+\\
+\uncover<6->{
+\Leftrightarrow\quad&\exists v\in V, v\ne 0, Av=\lambda v
+}
+\end{align*}
+\uncover<7->{%
+$\Rightarrow$ $\operatorname{Sp}A$ ist die Menge der Eigenwerte
+}
+\end{block}}
+\end{column}
+\begin{column}{0.48\textwidth}
+\uncover<8->{%
+\begin{block}{Unendlichdimensional}
+Es gibt eine Folge $x_n\in V$ von Einheitsvektoren
+$\|x_n\|=1$
+mit
+\begin{align*}
+\lim_{n\to\infty} (A - \lambda)x_n &= 0
+\end{align*}
+\end{block}}
+\uncover<9->{%
+\begin{block}{Spektrum und Norm}
+\[
+\operatorname{Sp}(A)
+\subset
+\{\lambda\in\mathbb{C}\;|\;
+|\lambda|\le \|A\|\}
+\]
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/stoneweierstrass.tex b/vorlesungen/slides/5/stoneweierstrass.tex
new file mode 100644
index 0000000..3f9cab5
--- /dev/null
+++ b/vorlesungen/slides/5/stoneweierstrass.tex
@@ -0,0 +1,11 @@
+%
+% stoneweierstrass.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\begin{frame}[t]
+\frametitle{Stone-Weierstrass}
+
+TODO XXX
+
+\end{frame}
diff --git a/vorlesungen/slides/5/unitaer.tex b/vorlesungen/slides/5/unitaer.tex
new file mode 100644
index 0000000..f0c4401
--- /dev/null
+++ b/vorlesungen/slides/5/unitaer.tex
@@ -0,0 +1,75 @@
+%
+% unitaer.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Unitäre Matrizen}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Eigenwerte}
+$U$ unitär lässt das Skalarprodukt invariant
+\[
+\langle Ux,Uy\rangle
+=
+\langle x,y\rangle
+\]
+\uncover<2->{%
+$\lambda$ ein Eigenwert mit Eigenvektor $v$:
+\begin{align*}
+\langle v,v\rangle
+&=
+\langle Uu,Uv\rangle
+\uncover<3->{= \langle \lambda v,\lambda v\rangle}
+\uncover<4->{= |\lambda|^2 \langle v,v\rangle}
+\\
+\uncover<5->{\Rightarrow\;|\lambda|&=1}
+\end{align*}}
+\end{block}
+\uncover<6->{%
+\begin{block}{Diagonalisierbar}
+Unitäre Matrizen sind über $\mathbb{C}$ diagonalisierbar
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Grosse Jordan-Blöcke?}
+Falls es Vektoren $v,w$ gibt mit
+\begin{align*}
+\uncover<7->{ Uv&=\lambda v}
+\\
+\uncover<8->{ Uw&=\lambda w + v}
+\intertext{\uncover<9->{Skalarprodukt:}}
+\uncover<10->{
+\langle v,w\rangle
+&=
+\langle Uv,Uw\rangle}
+\\
+\uncover<11->{
+&=
+\langle \lambda v,\lambda w\rangle
++
+\langle\lambda v,v\rangle}
+\\
+\uncover<12->{
+&=
+|\lambda|^2 \langle v,w\rangle
++
+\langle\lambda v,v\rangle}
+\\
+\uncover<13->{
+&=
+\langle v,w\rangle
++
+\lambda \| v\|^2}
+\\
+\uncover<14->{
+\Rightarrow\quad
+0&=\|v\|^2\quad\Rightarrow\quad \|v\|=0}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
diff --git a/vorlesungen/slides/5/verzerrung.tex b/vorlesungen/slides/5/verzerrung.tex
new file mode 100644
index 0000000..8d6514c
--- /dev/null
+++ b/vorlesungen/slides/5/verzerrung.tex
@@ -0,0 +1,121 @@
+%
+% verzerrung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\def\r{1.10}
+\def\s{1.12}
+\def\q{1.23}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Verzerrung}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.49\textwidth}
+\begin{block}{Abbildung $A\colon v\mapsto Av$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=2.5]
+\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1];
+
+\coordinate (a1) at (0.974,0.171);
+\coordinate (a2) at (0.037,1.018);
+
+\coordinate (v1) at (-0.5216,0.8532);
+\coordinate (v2) at (-0.3343,-0.9425);
+
+\foreach \a in {0,5,...,355}{
+	\draw[color=red,line width=1.2pt] 
+		($cos(\a)*(a1)+sin(\a)*(a2)$) --
+		($cos(\a+5)*(a1)+sin(\a+5)*(a2)$);
+}
+\foreach \a in {1,...,144}{
+	\only<\a>{
+		\fill[color=red,line width=1.4pt]
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03];
+		\draw[->,color=red,line width=1.4pt] (0,0) --
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$);
+		\draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1);
+		\fill[color=blue] ({5*\a}:1) circle[radius=0.03];
+		\node[color=blue] at ({5*\a}:\r) {$v$};
+		\node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$)
+			{$Av$};
+	}
+}
+
+\begin{scope}
+\clip (-1.2,-1.1) rectangle (1.2,1.1);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2);
+\end{scope}
+
+\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}];
+\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}];
+
+\node[color=darkgreen] at (v1) [above left] {$v_1$};
+\node[color=darkgreen] at (v2) [below left] {$v_2$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.49\textwidth}
+\uncover<73->{%
+\begin{block}{Abbildung $A\colon v\mapsto (A-\lambda)v$}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=2.5]
+\draw[color=blue,line width=1.2pt] (0,0) circle[radius=1];
+
+\coordinate (a1) at (0.121,0.343);
+\coordinate (a2) at (0.074,0.209);
+
+\coordinate (v1) at (-0.5216,0.8532);
+\coordinate (v2) at (-0.3343,-0.9425);
+
+\begin{scope}
+\clip (-1.2,-1.2) rectangle (1.2,1.2);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v1)$) -- ($2*(v1)$);
+\draw[color=darkgreen,line width=0.7pt] ($-2*(v2)$) -- ($2*(v2)$);
+\end{scope}
+
+\foreach \a in {0,5,...,355}{
+	\draw[color=red!60,line width=4pt] 
+		($cos(\a)*(a1)+sin(\a)*(a2)$) --
+		($cos(\a+5)*(a1)+sin(\a+5)*(a2)$);
+}
+\foreach \a in {73,...,144}{
+	\only<\a>{
+		\fill[color=red,line width=1.4pt]
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$) circle[radius=0.03];
+		\draw[->,color=red,line width=1.4pt] (0,0) --
+			($cos(\a*5)*(a1)+sin(\a*5)*(a2)$);
+		\draw[->,color=blue,line width=1.4pt] (0,0) -- ({5*\a}:1);
+		\fill[color=blue] ({5*\a}:1) circle[radius=0.03];
+		\node[color=blue] at ({5*\a}:\r) {$v$};
+		\node[color=red] at ($\s*cos(\a*5)*(a1)+\s*sin(\a*5)*(a2)$)
+			{$(A-\lambda)v$};
+	}
+}
+
+\begin{scope}
+\clip (-1.2,-1.1) rectangle (1.2,1.1);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v1);
+\draw[->,color=darkgreen,line width=1.5pt] (0,0) -- (v2);
+\end{scope}
+
+\draw[->] (-\q,0) -- (1.2,0) coordinate[label={$x$}];
+\draw[->] (0,-1.2) -- (0,1.2) coordinate[label={right:$y$}];
+
+\node[color=darkgreen] at (v1) [above left] {$v_1$};
+\node[color=darkgreen] at (v2) [below left] {$v_2$};
+
+\end{tikzpicture}
+\end{center}
+
+\end{block}}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
diff --git a/vorlesungen/slides/5/verzerrung/verzerrung.m b/vorlesungen/slides/5/verzerrung/verzerrung.m
new file mode 100644
index 0000000..028e7f9
--- /dev/null
+++ b/vorlesungen/slides/5/verzerrung/verzerrung.m
@@ -0,0 +1,13 @@
+#
+# verzerrung.m
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+
+rand("seed", 4712);
+
+A = eye(2) + 1.0 * (rand(2,2) - 0.5 * ones(2,2))
+
+[V, lambda] = eig(A)
+
+B = A - lambda(1,1) * eye(2)
diff --git a/vorlesungen/slides/5/zerlegung.tex b/vorlesungen/slides/5/zerlegung.tex
new file mode 100644
index 0000000..a734d69
--- /dev/null
+++ b/vorlesungen/slides/5/zerlegung.tex
@@ -0,0 +1,105 @@
+%
+% zerlegung.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{frame}[t]
+\frametitle{Zerlegung in Eigenräume}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-15pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{center}
+\begin{tikzpicture}[>=latex,thick,scale=0.38]
+\uncover<2->{
+\fill[color=blue!20] (0,11) rectangle (4,15);
+\fill[color=red!20] (4,0) rectangle (15,11);
+}
+\uncover<3->{
+\fill[color=red!40] (9,0) rectangle (15,6);
+\fill[color=blue!40,opacity=0.5] (4,6) rectangle (9,11);
+}
+\uncover<4->{
+\fill[color=blue!40,opacity=0.5] (9,3) rectangle (12,6);
+\fill[color=blue!40,opacity=0.5] (12,0) rectangle (15,3);
+}
+
+\uncover<2->{
+\draw[line width=0.1pt] (0,11) -- (15,11);
+\draw[line width=0.1pt] (4,0) -- (4,15);
+}
+
+\uncover<3->{
+\draw[line width=0.1pt] (0,6) -- (15,6);
+\draw[line width=0.1pt] (9,0) -- (9,15);
+}
+
+\uncover<4->{
+\draw[line width=0.1pt] (0,3) -- (15,3);
+\draw[line width=0.1pt] (12,0) -- (12,15);
+}
+\draw (0,0) rectangle (15,15);
+\uncover<2->{
+\node[color=darkgreen] at (2,15) [above] {$\mathcal{E}_{\lambda_1}$};
+\node[color=darkgreen] at (0,13) [above,rotate=90] {$\mathcal{K}(f-\lambda_1)$};
+\node at (2,13) {$f_{|\mathcal{E}_{\lambda_1}}$};
+}
+\uncover<3->{
+\node at (7,15) [above] {$\mathcal{E}_{\lambda_2}$};
+\node at (7,8.5) {$(f_1)_{|\mathcal{E}_{\lambda_2}}$};
+}
+\uncover<4->{
+\node at (10.5,15) [above] {$\mathcal{E}_{\lambda_3}$};
+\node at (13.5,15) [above] {$\mathcal{E}_{\lambda_4}$};
+\node at (10.5,4.5) {$(f_2)_{|\mathcal{E}_{\lambda_3}}$};
+}
+\end{tikzpicture}
+\end{center}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Iteration}
+$\Lambda=\{\lambda_1,\dots,\lambda_s\}$ Eigenwerte
+\begin{align*}
+\uncover<2->{
+V
+&=
+\mathcal{K}(f-\lambda_1)
+\oplus
+\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!20] (0,0) rectangle (1.83,1.1);}}}}
+\underbrace{\mathcal{J}(f-\lambda_1)}_{\displaystyle=V_1}
+}
+\\[-15pt]
+\uncover<2->{
+f_1 &= f_{|V_1}
+}
+\\[10pt]
+\uncover<3->{
+V_1
+&=
+\mathcal{K}(f_1-\lambda_2)
+\oplus
+\raisebox{-22pt}{\smash{\rlap{\tikz{\fill[color=red!40] (0,0) rectangle (1.9,1.1);}}}}
+\underbrace{\mathcal{J}(f_1-\lambda_2)}_{\displaystyle=V_2}
+}
+\\[-15pt]
+\uncover<3->{
+f_1 &= f_{|V_1}
+}
+\\
+\uncover<4->{
+&\phantom{0}\vdots
+}
+\end{align*}
+\uncover<5->{%
+$\Rightarrow$ $f$ hat {\color{blue}Blockdiagonalform} für die Zerlegung
+\begin{align*}
+V&=\bigoplus_{\lambda\in\Lambda} \mathcal{E}_{\lambda}
+\end{align*}}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup