BUILT-IN MATHEMATICA SYMBOL

# Eigensystem

Eigensystem[m]
gives a list 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.

## Details and OptionsDetails and Options

• 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 .
• 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].
• 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 and can be used to specify that explicit radicals should be generated for all cubics and quartics.
• SparseArray objects can be used in Eigensystem. »

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Symbolic eigenvalues and eigenvectors:

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Exact eigenvalues and eigenvectors:

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Numerical value:

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Eigenvalues and eigenvectors computed with numerical methods:

 Out[1]=