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Documentation / Mathematica / Built-in Functions / Lists and Matrices / Matrix Operations /

Eigenvalues

FilledSmallSquare Eigenvalues[m] gives a list of the eigenvalues of the square matrix m.

FilledSmallSquare Eigenvalues[m, a] gives the generalized eigenvalues of m with respect to a.

FilledSmallSquare Eigenvalues[m, k] gives the first k eigenvalues of m.

FilledSmallSquare Eigenvalues finds numerical eigenvalues if m contains approximate real or complex numbers.

FilledSmallSquare Repeated eigenvalues appear with their appropriate multiplicity.

FilledSmallSquare An matrix gives a list of exactly eigenvalues, not necessarily distinct.

FilledSmallSquare If they are numeric, eigenvalues are sorted in order of decreasing absolute value.

FilledSmallSquare The eigenvalues of a matrix m are those for which m . v == v for some non-zero eigenvector v.

FilledSmallSquare The generalized eigenvalues of m with respect to a are those for which m . v == a . v.

FilledSmallSquare When matrices m and a have a dimension- shared null space, then of their generalized eigenvalues will be Indeterminate.

FilledSmallSquare Ordinary eigenvalues are always finite; generalized eigenvalues can be infinite.

FilledSmallSquare For numeric eigenvalues, Eigenvalues[m, k] gives the k that are largest in absolute value.

FilledSmallSquare Eigenvalues[m, -k] gives the k that are smallest in absolute value.

FilledSmallSquare Eigenvalues[m, spec] is always equivalent to Take[Eigenvalues[m], spec].

FilledSmallSquare The option settings Cubics->True and Quartics->True can be used to specify that explicit radicals should be generated for all cubics and quartics.

FilledSmallSquare SparseArray objects can be used in Eigenvalues.

FilledSmallSquare See Section 1.8.3 and Section 3.7.9.

FilledSmallSquare See also: SingularValueList, CharacteristicPolynomial, Det, Tr.

FilledSmallSquare New in Version 1; modified in 5.0.

Further Examples