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 are those for which for some non-zero eigenvector .
  • The generalized eigenvalues of with respect to 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].
  • 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.

ExamplesExamplesopen allclose all

Basic Examples (6)Basic Examples (6)

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Exact eigenvalues:

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Find approximate numerical eigenvalues:

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Find eigenvalues starting with 20-digit precision:

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Largest 5 eigenvalues:

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Multiple eigenvalues are listed multiple times:

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