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#
# 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")
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