# Eigenvectors

Eigenvectors[m]

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

Eigenvectors[{m,a}]

gives the generalized eigenvectors of m with respect to a.

Eigenvectors[m,k]

gives the first k eigenvectors of m.

Eigenvectors[{m,a},k]

gives the first k generalized eigenvectors.

# Details and Options

• Eigenvectors finds numerical eigenvectors if m contains approximate real or complex numbers.
• For approximate numerical matrices m, the eigenvectors are normalized.
• For exact or symbolic matrices m, the eigenvectors are not normalized.
• Eigenvectors corresponding to degenerate eigenvalues are chosen to be linearly independent.
• For an nn matrix, Eigenvectors always returns a list of length n. The list contains each of the independent eigenvectors of the matrix, supplemented if necessary with an appropriate number of vectors of zeros. »
• Eigenvectors with numeric eigenvalues are sorted in order of decreasing absolute value of their eigenvalues.
• Eigenvectors[m,spec] is equivalent to Take[Eigenvectors[m],spec].
• Eigenvectors[m,UpTo[k]] gives k eigenvectors, or as many as are available.
• SparseArray objects can be used in Eigenvectors.
• 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 ZeroTest Automatic test to determine when expressions are zero
• The ZeroTest option only applies to exact and symbolic matrices.
• 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.

# Examples

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## Basic Examples(3)

Symbolic eigenvectors:

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Exact eigenvectors:

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Numerical value:

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Eigenvectors computed using numerical methods:

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# Tutorials

Introduced in 1988
(1.0)
| Updated in 2015
(10.3)