# Eigenvalues

Usage

Eigenvalues[m] gives a list of the eigenvalues of the square matrix m.
Eigenvalues[{m, a}] gives the generalized eigenvalues of m with respect to a.
Eigenvalues[m, k] gives the first k eigenvalues of m.

Notes

Eigenvalues finds numerical eigenvalues if m contains approximate real or complex numbers.
• Repeated eigenvalues appear with their appropriate multiplicity.
• An matrix gives a list of exactly eigenvalues, not necessarily distinct.
• If they are numeric, eigenvalues are sorted in order of decreasing absolute value.
• The eigenvalues of a matrix m are those for which m . v v for some non-zero eigenvector v.
• The generalized eigenvalues of m with respect to a are those for which m . v a . v.
• When matrices m and a have a dimension- shared null space, then of their generalized eigenvalues will be Indeterminate.
• Ordinary eigenvalues are always finite; generalized eigenvalues can be infinite.
• For numeric eigenvalues, Eigenvalues[m, k] gives the k that are largest in absolute value.
Eigenvalues[m, -k] gives the k that are smallest in absolute value.
Eigenvalues[m, spec] is always equivalent to Take[Eigenvalues[m], spec].
• The option settings Cubics->True and Quartics->True can be used to specify that explicit radicals should be generated for all cubics and quartics.
SparseArray objects can be used in Eigenvalues.
• New in Version 1; modified in 5.