From 23181dd294a16f4b30ce0531b23a4d0fc1abeca0 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Mon, 3 May 2021 20:53:09 +0200
Subject: new problem

---
 .../40-eigenwerte/uebungsaufgaben/4006.maxima      | 121 +++++++++++++++++++++
 1 file changed, 121 insertions(+)
 create mode 100644 buch/chapters/40-eigenwerte/uebungsaufgaben/4006.maxima

(limited to 'buch/chapters/40-eigenwerte/uebungsaufgaben/4006.maxima')

diff --git a/buch/chapters/40-eigenwerte/uebungsaufgaben/4006.maxima b/buch/chapters/40-eigenwerte/uebungsaufgaben/4006.maxima
new file mode 100644
index 0000000..9c97a2b
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/uebungsaufgaben/4006.maxima
@@ -0,0 +1,121 @@
+/*
+ * 4006.maxima
+ *
+ * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+
+A: matrix([ a+b*%i,      1,       0,      0 ],
+          [      0, a+b*%i,       0,      0 ],
+          [      0,      0,  a-b*%i,      1 ],
+          [      0,      0,       0, a-b*%i ]);
+
+expand(charpoly(A, x));
+
+S: (1/sqrt(2)) * matrix([ 1, -%i,  0,   0 ],
+                        [ 0,   0,  1, -%i ],
+                        [ 1,  %i,  0,   0 ],
+                        [ 0,   0,  1,  %i ]);
+
+B: expand(invert(S).A.S);
+
+
+C: subst(2, a, B);
+C: subst(3, b, C);
+A: subst(2, a, A);
+A: subst(3, b, A);
+
+U: matrix([ 1, 0, 1, 0 ],
+          [ 0, 1, 1, 2 ],
+          [ 0, 0, 1, 0 ],
+          [ 0, 0, 0, 1 ]);
+V: matrix([ 1, 0, 0, 0 ],
+          [ 0, 1, 0, 0 ],
+          [ 0, 1, 1, 0 ],
+          [ 1, 0, 0, 1 ]);
+T: U.V;
+invert(T);
+
+D: T.C.invert(T);
+
+p: expand(charpoly(D, x));
+
+factor(p);
+
+lambda: 2+3*%i;
+
+Dlambda: ratsimp(expand(D - lambda * identfor(D)));
+rank(Dlambda);
+/* D2: expand(Dlambda.Dlambda); */
+/* rank(D2); */
+
+load(functs);
+
+/*
+E: Dlambda;
+E[1]: (rational(1/E[1,1]))*E[1]$
+E[2]: E[2] - E[2,1] * E[1]$
+E[3]: E[3] - E[3,1] * E[1]$
+E[4]: E[4] - E[4,1] * E[1]$
+E: ratsimp(E)$
+
+E[2]: (rational(1/E[2,2])) * E[2]$
+E[3]: E[3] - E[3,2] * E[2]$
+E[4]: E[4] - E[4,2] * E[2]$
+E: ratsimp(E)$
+
+E[3]: (rational(1/E[3,3])) * E[3]$
+E[4]: E[4] - E[4,3] * E[3]$
+E: ratsimp(E)$
+
+E[2]: E[2] - E[2,3] * E[3]$
+E[1]: E[1] - E[1,3] * E[3]$
+E: ratsimp(E)$
+
+E[1]: E[1] - E[1,2] * E[2]$
+E: ratsimp(E)$
+
+E;
+*/
+
+b1: matrix([1+%i],[2+2*%i],[%i],[1]);
+ratsimp(D.b1 - lambda*b1);
+
+G: Dlambda;
+G: addcol(G, b1);
+G[1]: (rational(1/G[1,1]))*G[1]$
+G[2]: G[2] - G[2,1] * G[1]$
+G[3]: G[3] - G[3,1] * G[1]$
+G[4]: G[4] - G[4,1] * G[1]$
+G: ratsimp(G)$
+
+G[2]: (rational(1/G[2,2])) * G[2]$
+G[3]: G[3] - G[3,2] * G[2]$
+G[4]: G[4] - G[4,2] * G[2]$
+G: ratsimp(G)$
+
+G[3]: (rational(1/G[3,3])) * G[3]$
+G[4]: G[4] - G[4,3] * G[3]$
+G: ratsimp(G)$
+
+G[2]: G[2] - G[2,3] * G[3]$
+G[1]: G[1] - G[1,3] * G[3]$
+G: ratsimp(G)$
+
+G[1]: G[1] - G[1,2] * G[2]$
+G: ratsimp(G)$
+
+G;
+
+b2: matrix([ G[1,5] ], [ G[2,5] ], [ G[3,5] ], [ G[4,5] ]);
+
+expand(D.b2 - lambda * b2 - b1);
+
+c1: 2 * realpart(b1);
+d1: 2 * imagpart(b1);
+c2: 2 * realpart(b2);
+d2: 2 * imagpart(b2);
+
+D.c1 - 2 * c1 + 3 * d1;
+D.d1 - 3 * c1 - 2 * d1;
+D.c2 - 2 * c2 + 3 * d2 - c1;
+D.d2 - 3 * c2 - 2 * d2 - d1;
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