eig

Eigenvalues and eigenvectors of symbolic matrix

Syntax

lambda = eig(A)

[V,D] = eig(A)

[V,D,P] = eig(A)

lambda = eig(vpa(A))

[V,D] = eig(vpa(A))

Description

lambda = eig(A) returns a symbolic vector containing the eigenvalues of the square symbolic matrix A.

[V,D] = eig(A) returns matrices V and D. The columns of V present eigenvectors of A. The diagonal matrix D contains eigenvalues. If the resulting V has the same size as A, the matrix A has a full set of linearly independent eigenvectors that satisfy A*V = V*D.

[V,D,P] = eig(A) returns a vector of indices P. The length of P equals to the total number of linearly independent eigenvectors, so that A*V = V*D(P,P).

lambda = eig(vpa(A)) returns numeric eigenvalues using variable-precision arithmetic.

[V,D] = eig(vpa(A)) also returns numeric eigenvectors.

Examples

Compute Eigenvalues

Compute eigenvalues for the magic square of order 5.

M = sym(magic(5));
eig(M)

ans =
                                65
  (625/2 - (5*3145^(1/2))/2)^(1/2)
  ((5*3145^(1/2))/2 + 625/2)^(1/2)
 -(625/2 - (5*3145^(1/2))/2)^(1/2)
 -((5*3145^(1/2))/2 + 625/2)^(1/2)

Compute Numeric Eigenvalues to High Precision

Compute numeric eigenvalues for the magic square of order 5 using variable-precision arithmetic.

M = magic(sym(5));
eig(vpa(M))

ans =
                                65.0
 21.27676547147379553062642669797423
 13.12628093070921880252564308594914
  -13.126280930709218802525643085949
  -21.276765471473795530626426697974

Compute Eigenvalues and Eigenvectors

Compute the eigenvalues and eigenvectors for one of the MATLAB^® test matrices.

A = sym(gallery(5))

A =
[   -9,    11,   -21,     63,   -252]
[   70,   -69,   141,   -421,   1684]
[ -575,   575, -1149,   3451, -13801]
[ 3891, -3891,  7782, -23345,  93365]
[ 1024, -1024,  2048,  -6144,  24572]

[v, lambda] = eig(A)

v =
       0
  21/256
 -71/128
 973/256
       1
 
lambda =
[ 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0]
[ 0, 0, 0, 0, 0]

Input Arguments

A — Matrix

symbolic matrix

Matrix, specified as a symbolic matrix.

Limitations

Matrix computations involving many symbolic variables can be slow. To increase the computational speed, reduce the number of symbolic variables by substituting the given values for some variables.