Eigenvalues

gives a list of the eigenvalues of the square matrix m.

gives the generalized eigenvalues of m with respect to a.

gives the first k eigenvalues of m.

gives the first k generalized eigenvalues.

Details 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.