08c848a3-cec3-417f-83cd-3bd077598685 34 4 5 decimal

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Function for calculating the coefficients of the characteristic polynomial, eigenvalues and eigenvectors

Adapted from: "Introduction to the use of SMath Studio" G. Urroz, May 2010

A eig n A rows : p n 1 matrix : I n identity : p n 1 + 1 el 1.0 - : B A : p n 1 el B tr : j n 1 - 1 n 2 - range B A B p j 1 + 1 el I * - (* : p j 1 el B tr n j - 1 + / : 21 line for p 1 - (n^p * : λ p polyroots : V n 1 - 1 matrix : j 1 n range A.L A λ j el I * - : A.Lm A.L 1 n 1 - 1 n 1 - submatrix : b.L A.L 1 n 1 - n n submatrix : x.L A.Lm 1 -^b.L * : V x.L V augment : 51 line for V V 1 n 1 - 1 n submatrix : V.b 1 n matrix : j 1 n range V.b 1 j el 1 : for V V V.b stack : j 1 n range x.L V j col : x.L x.L x.L norme / : V V x.L augment : 31 line for V V 1 n n 1 + 2 n * submatrix : p λ V 31 sys 181 line :

12 - 422 - 22 - 642 - 38268444 mat eig 1300 - 530 - 93101 - 51 mat 1.95 - 6.33 7.83 - 13.45 41 mat 0.35 - 0.1 0.79 0.7 - 0.21 - 0.51 0.52 - 0.46 0.59 - 0.57 0.26 0.54 - 0.69 0.64 0.18 0.11 44 mat 31 sys

2 - i + 0002 - i - 0001 - 33 mat eig 595141 mat 1 - 2 - i + 2 - i - 31 mat 00000011133 mat 31 sys

EIGENVECTORS CALCULATION FAILS FOR 2x2 MATRICES:

1 - 004 - 22 mat eig 1 el 4 - 5 - 1 - 31 mat

1 - 004 - 22 mat eig 2 el 1 - 4 - 21 mat

1 - i - 001 - i + 22 mat eig 1 el 2 - 2 - 1 - 31 mat

1 - i - 001 - i + 22 mat eig 2 el 1 - i + 1 - i - 21 mat

1 - 004 - 22 mat eig 3 el 001122 mat

A 12 - 63010251 - 33 mat :

A eig 657 - 0141 mat 0.105 7.497 7.602 - 31 mat 0.554 0.244 - 0.873 0.574 0.81 - 0.408 - 0.603 0.534 0.266 33 mat 31 sys

1 - i - 001 - i + 22 mat eig 3 el 001122 mat

eigenvals A eig 2 el : 0.1053 7.4966 7.6019 - 31 mat

eigenvecs A eig 3 el : 0.554 0.2438 - 0.8732 0.5743 0.8097 - 0.4081 - 0.6028 0.5339 0.2663 33 mat

Each col of eigenvecs is the eigen vector corresponding to each row of eigenvals.

Does NOT sanity MCD 11

Sanity MCD 11 wrt col(eigenvecs,#)

Sanity MCD 11

A eigenvecs 3 col * 3.2871 5.2823 0.5606 - 31 mat

eigenvals 1 el eigenvecs 3 col * 0.0919 0.043 - 0.028 31 mat

A 0.8732 0.4081 - 0.2663 - 31 mat * 0.0916 0.0434 - 0.0278 - 31 mat

A eigenvecs 2 col * 4.5787 4.6072 5.0698 - 31 mat

eigenvals 2 el eigenvecs 2 col * 1.8275 - 6.0698 - 4.0021 31 mat

A 0.2438 0.8097 0.5339 31 mat * 1.8278 6.0704 4.0022 31 mat

A eigenvecs 1 col * 3.0219 7.6895 3.3766 31 mat

eigenvals 3 el eigenvecs 1 col * 4.2112 - 4.3657 - 4.5821 - 31 mat

A 0.554 - 0.5743 - 0.6028 31 mat * 4.2114 4.366 4.5823 - 31 mat

A dn_LinAlgEigenvalues 7.4966 0.1053 7.6019 - 31 mat

0.1053 7.4966 7.6019 - 31 mat

A dn_LinAlgEigenvectors 0.2438 0.8732 0.554 - 0.8097 0.4081 - 0.5743 - 0.5339 0.2663 - 0.6028 33 mat

0.554 0.2438 - 0.8732 0.5743 0.8097 - 0.4081 - 0.6028 0.5339 0.2663 33 mat

A norm1 17

A norme 13.4164

A normi 13