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].
  • SparseArray objects can be used in Eigenvalues.
  • Eigenvectors has the following options and settings:
  • CubicsFalsewhether to use radicals to solve cubics
    MethodAutomaticmethod to use
    QuarticsFalsewhether 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 method is also known as a Lanczos method when applied to symmetric or Hermitian matrices.
  • The and methods take suboptions Method->{"name",opt1->val1,}, which can be found in the Method subsection.
Introduced in 1988
(1.0)
| Updated in 2014
(10.0)