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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[{m, a}, k]
gives the first k generalized eigenvalues and eigenvectors.

MORE INFORMATION

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

For approximate numerical matrices m, the eigenvectors are normalized.

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 applying Take[..., spec] to each element of Eigensystem[m].

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

SparseArray objects can be used in Eigensystem. »

Basic Examples (3)

Symbolic eigenvalues and eigenvectors:

Exact eigenvalues and eigenvectors:

Numerical value:

Eigenvalues and eigenvectors computed with numerical methods:

Generalizations & Extensions (2)

Applications (2)

Properties & Relations (5)

Possible Issues (3)

Eigenvalues Eigenvectors NullSpace JordanDecomposition SchurDecomposition SingularValueDecomposition QRDecomposition

Eigenvalues and Eigenvectors

Matrix Decompositions

Matrix Operations

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