<p> eigenall </p>

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This document can be used two ways: 1. Solving for eigenvalues and eigenvectors, given a square matrix. 2. Create the "companion matrix", which you can feed into "eigenall" Note that this 'return' is via 'polyroots' [built-in Smath]. Observe that 'eigenall' applies to any sparse square matrix, whatever your matrix may represent. But if you want the eigenvectors wrt your polynomial, you MUST provide the companion matrix. -----------------------More interesting --------------------------- Assume it is not easy to transmit a polynomial by text only, then create the 'companion matrix' and transmit [may be as a matrix or as unwrapped vector], your receiver will then be able to recreate your polynomial via the first solution of the 'eigenall'. As well, your receiver will enjoy the eigenvalues, i.e: the roots of the polynomial & the eigenvectors.

x P 42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+ : Lagrange canonical ≡

The Smath 5346 Maple wraper works a bit different than the Mathcad 11/Maple 11. You feed the RHS of P(x) as the argument to maple(op( ,,,,))

x P

42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+

x expr 42 - 27 x * - 13 x 2^* + 5 x 3^* - x 4^+ op maple :

x 1 :

coeffs x expr :

# op maple is undocumented in Mathcad 11 and earlier ≡

coeffs 42 - 27 - 135 - 151 mat

vx companion c vx : a 0 : leading c c rows el : i 1 c length range a i el c i el - leading / : for b a 1 c length 1 - 11 submatrix : j 1 c length 2 - range M j 1 + j el 1 : for M b augment 71 line :

E coeffs companion :

E 000421002701013 - 001544 mat

x P 0 2 1 sys

polyroots is Smath built-in coeffs polyroots 0.9429 - 0.9402 3.1749 i * + 0.9402 3.1749 i * - 4.0624 41 mat

<p> eigenall </p>

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 1.3 103^* - 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 5 - 9 - 5 - 1 - 41 mat 1 - 2 - i + 2 - i - 31 mat 00000011133 mat 31 sys

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

ε 106 -^:

EIGENVECTORS CALCULATION FAILS FOR 2x2 MATRICES:

1 - ε + 0 ε + 0 ε - 4 - 22 mat eig 3 el 3.33 107 -^* 11 3.35 107 -^* 22 mat

1 - 004 - 22 mat eig 1 el 45131 mat

1 - i - 001 - i + 22 mat eig 1 el 22131 mat

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

E 000421002701013 - 001544 mat :

E eig 42 - 27 - 135 - 151 mat 0.9429 - 0.9402 3.1749 i * + 0.9402 3.1749 i * - 4.0624 41 mat 0.7437 - 0.2125 - 0.7175 i * - 0.2125 - 0.7175 i * + 0.9156 0.6612 - 0.0529 0.5843 i * - 0.0529 0.5843 i * + 0.3824 - 0.0674 0.2395 0.1873 i * + 0.2395 0.1873 i * - 0.1222 0.0719 0.059 0.059 0.0206 44 mat 31 sys

coeffs E eig 1 el : 42 - 27 - 135 - 151 mat

roots E eig 2 el : 0.942929 - 0.940248 3.17495 i * + 0.940248 3.17495 i * - 4.062433 41 mat

1 - 004 - 22 mat eig 3 el 001122 mat

M E eig 3 el :

eigvec M 1 col : 0.74372 - 0.661178 - 0.067445 0.071936 41 mat

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

eigval M 1 M rows 2 M cols submatrix : 0.212474 - 0.717463 i * - 0.212474 - 0.717463 i * + 0.91564 0.052945 0.584278 i * - 0.052945 0.584278 i * + 0.382433 - 0.239495 0.187298 i * + 0.239495 0.187298 i * - 0.122167 0.058993 0.058993 0.020557 43 mat