CoCoA:Eigenvectors

Eigenvectors

eigenvalues and eigenvectors of a matrix

Description

M must be a matrix of numbers, and X an indeterminate.

This function determines the eigenvalues of M, and for each eigenvalue

gives a basis of the corresponding eigenspace -- note that the basis is

probably not orthogonal. For irrational eigenvalues, the minimal polynomial of the eigenvalue is given (as a polynomial in X), along with the eigenvectors expressed in terms of a root of the minimal polynomial (represented as X).

Example

  Use R ::= Q[x];
  M := Mat([[1,2,3],[4,5,6],[7,8,9]]);
  Eigenvectors(M, x);
[Record[Eigenspace = [[-1, 2, -1]], MinPoly = x],
Record[Eigenspace = [[1, 1/8x + 1/4, 1/4x - 1/2]], MinPoly = x^2 - 15x - 18]]
-------------------------------
  M := Mat([[0,2,0,0],[1,0,0,0],[0,0,0,2],[0,0,1,0]]);
  Eigenvectors(M, x); -- two irrational eigenvalues, each with eigenspace of dimension 2
[Record[Eigenspace = [[1, 1/2x, 0, 0], [0, 0, 1, 1/2x]], MinPoly = x^2 - 2]]
-------------------------------

Syntax

Eigenvectors(M:MAT, X:POLY):LIST

   <type>matrix</type>