# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# Eigenvalues

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.

Eigenvalues[{m,a},k]
gives the first k generalized eigenvalues.

## Details and OptionsDetails and Options

• 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 for some nonzero eigenvector .
• The generalized eigenvalues of m with respect to a are those for which .
• 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].
• Eigenvalues[m,UpTo[k]] gives k eigenvalues, or as many as are available.
• SparseArray objects can be used in Eigenvalues.
• Eigenvectors has the following options and settings:
•  Cubics False whether to use radicals to solve cubics Method Automatic method to use Quartics False whether to use radicals to solve quartics
• Explicit Method settings for approximate numeric matrices include:
•  "Arnoldi" Arnoldi iterative method for finding a few eigenvalues "Banded" direct banded matrix solver "Direct" direct method for finding all eigenvalues "FEAST" FEAST iterative method for finding eigenvalues in an interval (applies to Hermitian matrices only)
• The "Arnoldi" method is also known as a Lanczos method when applied to symmetric or Hermitian matrices.
• The "Arnoldi" and "FEAST" methods take suboptions Method->{"name",opt1->val1,}, which can be found in the Method subsection.

## ExamplesExamplesopen allclose all

### Basic Examples  (4)Basic Examples  (4)

Machine-precision numerical eigenvalues:

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Approximate 20-digit precision eigenvalues:

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Exact eigenvalues:

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Symbolic eigenvalues:

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