Eigensystem

Eigensystem[m] gives a list values, vectors of the eigenvalues and eigenvectors of the square matrix m.

Eigensystem[m, a] gives the generalized eigenvalues and eigenvectors of m with respect to a.

Eigensystem[m, k] gives the eigenvalues and eigenvectors for the first k eigenvalues of m.

Eigensystem finds numerical eigenvalues and eigenvectors if m contains approximate real or complex numbers.

All the non-zero eigenvectors given are independent. If the number of eigenvectors is equal to the number of non-zero eigenvalues, then corresponding eigenvalues and eigenvectors are given in corresponding positions in their respective lists.

If there are more eigenvalues than independent eigenvectors, then each extra eigenvalue is paired with a vector of zeros.

Eigensystem[m, ZeroTest -> test] applies test to determine whether expressions should be assumed to be zero. The default setting is ZeroTest -> Automatic.

The eigenvalues and eigenvectors satisfy the matrix equation m.Transpose[vectors] == Transpose[vectors].DiagonalMatrix[values].

Generalized eigenvalues and eigenvectors satisfy m.Transpose[vectors] == a.Transpose[vectors].DiagonalMatrix[values].

vals, vecs = Eigensystem[m] can be used to set vals and vecs to be the eigenvalues and eigenvectors respectively.

Eigensystem[m, spec] is equivalent to Take[Eigensystem[m], spec].

SparseArray objects can be used in Eigensystem.

See notes for Eigenvalues.

See Section 3.7.9.

See also: NullSpace, JordanDecomposition, SchurDecomposition, SingularValueDecomposition, QRDecomposition.

Related package: LinearAlgebra`Orthogonalization`.

New in Version 1; modified in 5.0.

Further Examples