Symbolic Math Toolbox

eig

Symbolic matrix eigenvalues and eigenvectors

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 a matrix V whose columns are eigenvectors and a diagonal matrix D containing eigenvalues. If the resulting V is the same size as A, then A has a full set of linearly independent eigenvectors that satisfy A*V = V*D.

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

lambda = eig(VPA(A)) and [V,D] = eig(VPA(A)) compute numeric eigenvalues and eigenvectors, respectively, using variable precision arithmetic. If A does not have a full set of eigenvectors, the columns of V will not be linearly independent.

Examples

The statements

• R = sym(gallery('rosser'));
  eig(R)
  

return

• ans =
  [                0]
  [             1020]
  [ 510+100*26^(1/2)]
  [ 510-100*26^(1/2)]
  [   10*10405^(1/2)]
  [  -10*10405^(1/2)]
  [             1000]
  [             1000]
  

eig(vpa(R)) returns

• ans =
  

• [    -1020.0490184299968238463137913055]
  [ .56512999999999999999999999999800e-28]
  [  .98048640721516997177589097485157e-1]
  [     1000.0000000000000000000000000002]
  [     1000.0000000000000000000000000003]
  [     1019.9019513592784830028224109024]
  [     1020.0000000000000000000000000003]
  [     1020.0490184299968238463137913055]
  

The statements

• A = sym(gallery(5));
  [v,lambda] = eig(A) 
  

return

• 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]
  

See Also

jordan, poly, svd, vpa

dsolve

expm