2.4 Eigenvalues and Eigenvectors

Given a quadratic matrix $\mbox{${\bf A}$}$, the eigenvalues $\lambda_{i}$ are defined by

$$\left\vert \mbox{${\bf A}$}-\lambda_{i} \, \mbox{${\bf I}$} \right\vert = 0 \;\;\;\; (i=1, \dots N)$$ (2.3)

and the corresponding eigenvectors $\vec{a}_{i}$ are

$$\left[ \mbox{${\bf A}$} - \lambda_{i} \, \mbox{${\bf I}$}\, \right] \cdot \vec{a}_{i} = 0 \, ,$$ (2.4)

Normally one uses library subroutines, such as NAG F01xxx or ESSL SGEEV.

If only a few eigenvalues and the associated eigenvectors are needed then simple iterative procedures may be used.