a91d4bd0-498f-40e5-a29e-bd9bd8e00a76 43 6 6 decimal

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The companion matrix =>

companion matrix dn_LinAlgEigenvalues

This work sheet will help construct the vector of coefficients required by "polyroots". The "companion matrix" provides the argument to dn_LinAlgEigenvalues(companion matrix). You may need the "companion matrix" or "coeffs" for other math, for instance spooling little piece of information, transmit to community members for recovery at home.

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 LHS of P(x) as the argument to maple(op( ,,,,))

x P

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

Undocumented

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

x 1 :

coeffs x expr :

The companion matrix, argument to dn_LinAlgEigenvalues(E) 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 :

coeffs transpose 42 - 27 - 135 - 115 mat

E coeffs companion :

E 000421002701013 - 001544 mat

polyroots E dn_LinAlgEigenvalues ≡

r coeffs polyroots :

E dn_LinAlgEigenvalues 0.943 - 0.94 3.175 i * + 0.94 3.175 i * - 4.062 41 mat

x Lagrange x r 1 el - (x r 2 el - (* x r 3 el - (* x r 4 el - (* Re eval :

Lagrange complete plot, clean of residual imaginary

x Lagrange 0 r 1 el 0 . 14 black 1 5 mat r 4 el 0 . 14 black 1 5 mat 4 1 sys

E dn_LinAlgEigenvectors 0.9 0.7 - 0.7 - 0.7 0.4 - 0.5 - 0.2 i * + 0.5 - 0.2 i * - 0.7 0.1 0.2 0.2 i * + 0.2 0.2 i * - 0.1 - 0 0.1 i * - 0.1 i * 0.1 44 mat

Singular value Decompostion

A 22 - 151423 mat :

A dn_LinAlgSVD 6.775117 000 2.25783 023 mat 0.309244 - 0.950983 - 0.950983 - 0.309244 22 mat 0.793108 - 0.049076 - 0.6071 - 0.15756 - 0.979352 0.126668 0.588348 - 0.196116 - 0.784465 33 mat 31 mat

V A A transpose * : 912124222 mat

λ V dn_LinAlgEigenvalues : 5.097794 45.902206 21 mat

V dn_LinAlgEigenvectors 0.950983 - 0.309244 - 0.309244 0.950983 - 22 mat

v λ sqrt mat2sys sys2mat : 2.25783 6.775117 21 mat

v Diag 2.2578 00 6.7751 22 mat

Some graphic representation

2.2578 0 0 6.7751 2 2 mat 0.950983 - 0.309244 - 0.309244 0.950983 - 2 2 mat 2 1 sys V

Working with rref

A 0.3 0.2 6.6 1.1 - 10 4.5 1.8 - 0.3 - 6.5 5 7.3 - 9.7 10.9 4.1 - 2 8.1 2.7 - 8.7 8.9 6 - 45 mat :

A rref maple 1000 70.514516 - 0100 51.467536 - 0010 12.263383 0001 35.900426 45 mat

M 12124228133 mat :

rref M rref maple : 1032 / 0114 / - 00033 mat

A 317911 - 11724 mat :

rref A rref maple :

rref 102 6.5 011 10.5 - 24 mat

x 2 z * - .5 + ≡ y z - 10.5 - ≡ 21 mat

A 121 - 2.3 442 - 5.1 10 0.8 643 mat :

A rref maple 10001000100043 mat