# eigenvectors -- find eigenvectors of a matrix over RR or CC

## Synopsis

• Usage:
(eigvals, eigvecs) = eigenvectors M
• Inputs:
• Optional inputs:
• Hermitian => ..., default value false, Hermitian=>true means assume the matrix is symmetric or Hermitian
• Outputs:
• eigvals, , a list of the eigenvalues of M
• eigvecs, , or , if M is one), whose columns are the corresponding eigenvectors of M

## Description

The resulting matrix is over CC, and contains the eigenvectors of M. The lapack library is used to compute eigenvectors of real and complex matrices.

Recall that if v is a non-zero vector such that Mv = av, for a scalar a, then v is called an eigenvector corresponding to the eigenvalue a.

 i1 : M = matrix{{1, 2}, {5, 7}} o1 = | 1 2 | | 5 7 | 2 2 o1 : Matrix ZZ <--- ZZ i2 : eigenvectors M o2 = ({-.358899}, | -.827138 -.262266 |) {8.3589 } | .561999 -.964996 | o2 : Sequence
If the matrix is symmetric (over RR) or Hermitian (over CC), this information should be provided as an optional argument Hermitian=>true. In this case, the resulting eigenvalues will be returned as real numbers, and if M is real, the matrix of eigenvectors will be real.
 i3 : M = matrix {{1, 2}, {2, 1}} o3 = | 1 2 | | 2 1 | 2 2 o3 : Matrix ZZ <--- ZZ i4 : (e,v) = eigenvectors(M, Hermitian=>true) o4 = ({-1}, | -.707107 .707107 |) {3 } | .707107 .707107 | o4 : Sequence i5 : class \ e o5 = {RR} {RR} o5 : VerticalList i6 : v o6 = | -.707107 .707107 | | .707107 .707107 | 2 2 o6 : Matrix RR <--- RR 53 53

## Caveat

The eigenvectors are approximate.