3.7.9 Eigenvalues and Eigenvectors

Eigenvalues[m]	a list of the eigenvalues of m
Eigenvectors[m]	a list of the eigenvectors of m
Eigensystem[m]	a list of the form {eigenvalues, eigenvectors}
Eigenvalues[ N[m] ], etc.	numerical eigenvalues
Eigenvalues[ N[m, p] ], etc.	numerical eigenvalues, starting with -digit precision
CharacteristicPolynomial[m, x]	the characteristic polynomial of m

The eigenvalues of a matrix are the values for which one can find non-zero vectors such that . The eigenvectors are the vectors .

The characteristic polynomial CharacteristicPolynomial[m, x] for an matrix is given by Det[m - x IdentityMatrix[n]]. The eigenvalues are the roots of this polynomial.

Finding the eigenvalues of an matrix in general 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.

Here is a numerical matrix.

In[2]:=  m = {{2.3, 4.5}, {6.7, -1.2}} Out[2]=

This finds the eigenvalues of a random matrix. For non-symmetric matrices, the eigenvalues can have imaginary parts.

In[7]:=  Eigenvalues[ Table[Random[ ], {4}, {4}] ] Out[7]=

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 .

Here is a matrix.

In[8]:=  mz = {{0, 1, 0}, {0, 0, 1}, {0, 0, 0}} Out[8]=

Eigenvalues[m, k]	the largest k eigenvalues of m
Eigenvectors[m, k]	the corresponding eigenvectors of m
Eigenvalues[m, -k]	the smallest k eigenvalues of m
Eigenvectors[m, -k]	the corresponding eigenvectors of m

Eigenvalues sorts numeric eigenvalues so that the ones with large absolute value come first. In many situations, you may be interested only in the largest or smallest eigenvalues of a matrix. You can get these efficiently using Eigenvalues[m, k] and Eigenvalues[m, -k].

Eigenvalues[{m, a}]	the generalized eigenvalues of m with respect to a
Eigenvectors[{m, a}]	the generalized eigenvectors of m
CharacteristicPolynomial[{m, a}, x]
	the generalized characteristic polynomial of m

The generalized eigenvalues for a matrix with respect to a matrix are defined to be those for which .

The generalized eigenvalues correspond to zeros of the generalized characteristic polynomial Det[m - x a].

Note that while ordinary matrix eigenvalues always have definite values, some generalized eigenvalues will always be Indeterminate if the generalized characteristic polynomial vanishes, which happens if and share a null space. Note also that generalized eigenvalues can be infinite.