## 3.7.9 Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors.

The eigenvalues of a matrix are the values for which one can find non-zero vectors such that . The eigenvectors are the vectors .
Finding the eigenvalues of an matrix in principle involves solving an degree polynomial equation. For , therefore, the results cannot in general be expressed purely in terms of explicit radicals. Root objects can nevertheless always be used, although except for fairly sparse or otherwise simple matrices the expressions obtained are often unmanageably complex.

If you give a matrix of approximate real numbers, Mathematica will find the approximate numerical eigenvalues and eigenvectors.

The function Eigenvalues always gives you a list of eigenvalues for an matrix. The eigenvalues correspond to the roots of the characteristic polynomial for the matrix, and may not necessarily be distinct. Eigenvectors, on the other hand, gives a list of eigenvectors which are guaranteed to be independent. If the number of such eigenvectors is less than , then Eigenvectors appends zero vectors to the list it returns, so that the total length of the list is always .