plot(data,char,size,clr)

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&[DATE] &[TIME] - &[FILENAME]

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

eigenall Smath Designers

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

Companion Matrix

Companion Matrix ... sanity Mathcad 11 vx companion M 0 : 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 81 line :

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

x P x P op maple :

x 1 :

coeffs x P : 42 - 27 - 135 - 151 mat

coeffs coeffs eval :

E coeffs companion : 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

◄ characteristic polynomial

◄ char-poly roots

Each col < M > is ▲ the eignen vector ... of the corresponding row-roots

canonical Lagrange E eig 1 el transpose 42 - 27 - 135 - 115 mat

roots E eig 2 el : 0.9429 - 0.9402 3.1749 i * + 0.9402 3.1749 i * - 4.0624 41 mat

r roots :

Each col < M > is ▲ the eignen vector 
... of the corresponding row-roots M E eig 3 el :

commutative if exploded [index 1] roots 1 row 1 el M 1 col * 0.7013 0.6234 0.0636 - 0.0678 - 41 mat

M 1 col roots 1 row * 0.7013 0.6234 0.0636 - 0.0678 - 41 mat

E M 1 col * 3.0213 1.1986 1.5963 - 0.4271 41 mat

x f x r n el - (n 1 r rows product :

x c 1 1.5 - x ≤ (x 4.5 ≤ (& cases :

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

Independent complete plots

x f Re x c * x P 2 1 sys

=====================================================

a 204 - 6114 mat transpose :

x L a 1 el a 2 el x * + a 3 el x 2^* + a 4 el x 3^* + : (Lagrange canonic format ≡

r a polyroots :

r 0.4915 1.6194 i * + 0.4915 1.6194 i * - 6.983 - 31 mat

x f x r n el - (n 13 product :

x c 15 - x ≤ (x 3 ≤ (& cases :

Independent complete plots

x f Re x c * x L 2 1 sys

polyroots("vector") : Returns the roots of the polynomial whose coefficients are in "vector"

a1
a2
a3 r 1 el r 2 el * r 3 el * abs r 1 el r 2 el * r 1 el r 3 el * + r 2 el r 3 el * + r 1 el r 2 el + r 3 el + (- 31 mat 204 - 631 mat

Remarkable properties of the complex cubic

=====================================================

x P 42 - 27 x * - 13 x 2^* + 5 x 3^* - :

install  the ...
companion vector v 42 - 27 - 135 - 41 mat :

x P 0 2 1 sys

polyroots is Smath built-in r v polyroots : 0.9551 - 1.7775 2.374 i * + 1.7775 2.374 i * - 31 mat

ON/OFF to see effect plot r r reverse :

x Φ x r n el - (n 13 product eval :

x c 12 - x ≤ (x 2 ≤ (& cases :

this Polyroots needs be scaled ! scale v 4 el :

reconstruct from polyroots

scale x Φ * x c * x P 2 1 sys

v 42 - 27 - 135 - 41 mat

=====================================================

E v companion :

The companion matrix E 00 8.4 - 10 5.4 - 01 2.6 33 mat

install the companion matrix
... in 'eigen' system ... E 00 8.4 - 10 5.4 - 01 2.6 33 mat :

E eig 8.4 - 5.4 - 2.6 1 - 41 mat 0.9551 - 1.7775 2.374 i * + 1.7775 2.374 i * - 31 mat 0.4335 0.579 i * + 0.4335 0.579 i * - 0.922 - 0.21 0.6062 i * + 0.21 0.6062 i * - 0.3727 0.2554 0.2554 0.1048 33 mat 31 sys

v 42 - 27 - 135 - 41 mat :

v polyroots 0.9551 - 1.7775 2.374 i * + 1.7775 2.374 i * - 31 mat

roots E eig 2 el : 0.955057 - 1.777529 2.373958 i * + 1.777529 2.373958 i * - 31 mat

M E eig 3 el :

x m 42 - 27 x * - 13 x 2^* + 5 x 3^* - :

x Q v n el x n 1 -^* n 1 v rows sum :

x Clear 1

x Q x c * x m 2 1 sys

clear otherwise silent unit x x :

x Q expand maple 42 - 27 x * - 13 x 2^* + 5 x 3^* -

v 42 - 27 - 135 - 41 mat

eigen square matrix A 20003404933 mat :

A det 22

A eig 2235 - 141 - 41 mat 121131 mat 000 0.4472 - 0.9701 0.8944 0.8944 0.2425 0.4472 33 mat 31 sys

◄ characteristic polynomial

confusing A dn_LinAlgEigenvalues 111231 mat

◄ char-poly roots

A dn_LinAlgEigenvectors 001 0.4472 0.8944 0 0.8944 0.4472 - 033 mat

https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

M a b c d 22 mat :

λ 12 / a d + (* 12 / a d - (* (2^b c * + sqrt ± :

a 1 : b 2 : c 2 : d 1 : 22 mat

12 / a d + (* 12 / a d - (* (2^b c * + sqrt + 12 / a d + (* 12 / a d - (* (2^b c * + sqrt - 21 mat 31 - 21 mat

λ 12 / a d + (* 12 / a d - (* (2^b c * + sqrt ± : 31 - 21 sys

dn A dn_LinAlgEigenvectors eval :

v1 dn 1 col : v2 dn 2 col : v3 dn 3 col : 31 sys

v1 v3 augment v2 v3 augment 2 1 sys v1 v2 † v3 augment v1 v2 † v3 reverse augment 1.5 v3 reverse v1 v2 † augment ( * 3 1 sys

NOT commutative 
for this demo. v1 v2 † 1 - 0031 mat

M 1344493 - 43 - 5733 mat :

Vec M dn_LinAlgEigenvectors : 0.5386 - 0.8383 0.0845 0.8375 0.5436 0.0546 - 0.0917 0.0414 - 0.9949 33 mat

Vals M dn_LinAlgEigenvalues : 6.0992 15.3965 57.5043 31 mat

Vals Vals diag : 6.0992 000 15.3965 000 57.5043 33 mat

sanity Mathcad S Vec Vals sqrt vectorize * Vec transpose * : 3.5281 0.6391 0.3795 0.6391 2.9147 0.3103 - 0.3795 0.3103 - 7.5339 33 mat

sanity Mathcad S 2^1344493 - 43 - 5733 mat

X 10100032 mat :

Y 01010032 mat :

Z 001 0.5 0032 mat :

X Y Z 3 1 sys

m eigenvec m dn_LinAlgEigenvectors : m eigenvals m dn_LinAlgEigenvalues : 21 sys

Given a square matrix 'M'
a nonzero vector'v' is an eigenvector 
of 'M' if you can find  a number 'λ' 
such that M*v ≡ λ*v M 4213375104 - 33 mat :

sanity Mathcad 11 λ M eigenvals : 5.3718743743742 - 3.57717043464932 8.79470393972489 31 mat

sanity MCD 11
col<3> positive V M eigenvec : 0.807383254651 0.680299919996 - 0.467490164371 - 0.042069492339 0.727413953376 0.883242852422 - 0.588525647634 - 0.089782845175 - 0.036537786773 - 33 mat

index 1 to unest
the collection λ0 λ 1 row 1 el : λ1 λ 2 row 1 el : λ2 λ 3 row 1 el : 31 mat

v0 V 1 col : v1 V 2 col : v2 V 3 col : 13 mat

U0 λ0 v0 * : 4.3372 - 0.226 - 3.1615 31 mat

M v0 * 4.3372 - 0.226 - 3.1615 31 mat

U1 λ1 v1 * : 2.4335 - 2.6021 0.3212 - 31 mat

M v1 * 2.4335 - 2.6021 0.3212 - 31 mat

U2 λ2 v2 * : 4.1114 - 7.7679 - 0.3213 - 31 mat

M v2 * 4.1114 - 7.7679 - 0.3213 - 31 mat

Mathcad 11 . . .

The results returned by eigenvec and eigenvecs do not match, yet they are both valid solutions.

It is important to remember that an eigenvector is not a unique thing. For a given eigenvalue,

there are infinitely many eigenvectors.

The differences in the outputs of eigenvec and eigenvecs can be attributed to differences in their

underlying algortthms. The eigenvec function uses an inverse iteration algorithm, whereas

eigenvecs uses a reduction to Hessenberg form coupled with QR decomposition algorithm.

In summary, the guidelines for using eigenvec and eigenvecs are:

Use eigenvec if you know an eigenvalue for an array and wish to find a corresponding

eigenvector for that eigenvalue.

Use eigenvecs if you would like an eigenvector for each eigenvalue all at once or if you need

eigenvectors without computing any eigenvalues whatsoever.

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M 4213375104 - 33 mat :

M dn_LinAlgSVD 15.8746 000 5.799 000 1.8358 33 mat 0.8522 - 0.4099 - 0.3253 - 0.4798 - 0.8601 0.1734 0.2088 0.3038 0.9296 - 33 mat 0.2923 - 0.319 - 0.9016 - 0.2146 0.8968 0.3868 - 0.9319 - 0.3066 0.1936 33 mat 31 mat

Given an SVD ... retrieve the matrix

svd 3.464101615 000 3.16227766 023 mat 0.707106781 - 0.707106781 - 0.707106781 - 0.707106781 22 mat 0.40824829 - 0.816496581 - 0.40824829 - 0.894427191 - 0.447213595 0 0.182574186 - 0.365148372 - 0.912870929 33 mat 31 mat :

S svd 2 el : U svd 1 el : V svd 3 el : 13 mat

NOT commutative AUTO mode matrix S U * V * (: 3111 - 3123 mat

Given an SVD ... retrieve the matrix

M 4213375104 - 33 mat :

svd M dn_LinAlgSVD : 15.8746 000 5.799 000 1.8358 33 mat 0.8522 - 0.4099 - 0.3253 - 0.4798 - 0.8601 0.1734 0.2088 0.3038 0.9296 - 33 mat 0.2923 - 0.319 - 0.9016 - 0.2146 0.8968 0.3868 - 0.9319 - 0.3066 0.1936 33 mat 31 mat

S svd 2 el : U svd 1 el : V svd 3 el : 13 mat

NOT commutative AUTO mode matrix S U * V * (: 4213375104 - 33 mat

Polyroots Mathcad Laguerre differs slightly vs Companion Matrix Smath Polyroots differs from Mathcad 11... Not same source code. Mathcad 11 solves polyroots of complex vector ... NOT Smath 6179.