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Diffstat (limited to 'vorlesungen/slides/5')
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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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 Binary files differnew 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 |
