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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.
  • Eigenvalues finds numerical eigenvalues if m contains approximate real or complex numbers.
  • Repeated eigenvalues appear with their appropriate multiplicity.
  • An n×n matrix gives a list of exactly n 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 lambda for which m.vlambda v for some non-zero eigenvector v.
  • The generalized eigenvalues of m with respect to a are those lambda for which m.vlambda a.v.
  • When matrices m and a have a dimension-d shared null space, then d 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.
  • The option settings Cubics->True and Quartics->True can be used to specify that explicit radicals should be generated for all cubics and quartics.
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