This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Eigenvectors

 Eigenvectors[m]gives a list of the eigenvectors of the square matrix m. Eigenvectorsgives the generalized eigenvectors of m with respect to a. Eigenvectorsgives the first k eigenvectors of m. Eigenvectorsgives the first k generalized eigenvectors.
• Eigenvectors finds numerical eigenvectors if m contains approximate real or complex numbers.
• For approximate numerical matrices m, the eigenvectors are 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, followed if necessary by an appropriate number of vectors of zeros. »
• Eigenvectors with numeric eigenvalues are sorted in order of decreasing absolute value of their eigenvalues.
• Eigenvectors[m, ZeroTest->test] applies test to determine whether expressions should be assumed to be zero. The default setting is ZeroTest.
• The option settings Cubics->True and Quartics->True can be used to specify that explicit radicals should be generated for all cubics and quartics.
Symbolic eigenvectors:
Exact eigenvectors:
Numerical value:
Eigenvectors computed using numerical methods:
Symbolic eigenvectors:
 Out[1]=

Exact eigenvectors:
 Out[1]=
Numerical value:
 Out[2]=

Eigenvectors computed using numerical methods:
 Out[1]=
 Scope   (4)
Eigenvectors computed using numerical methods with 20-digit precision:
A Hilbert matrix:
Eigenvectors corresponding to the eigenvalues with smallest magnitude:
The corresponding eigenvalues:
Zero vectors are used when there are more eigenvalues than independent eigenvectors:
Compute generalized eigenvectors:
A block diagonal matrix:
Compute the eigenvectors:
A tridiagonal matrix:
The eigenvectors corresponding to the three largest eigenvalues:
 Applications   (2)
The eigenvectors of a 3×3 matrix m:
Diagonalize m:
The eigenvalues of a nondiagonalizable matrix:
Find the dimension of the span of all the eigenvectors:
Estimate the probability that a random 4×4 matrix of ones and zeros is not diagonalizable:
Compute the eigenvectors for a random symmetric matrix:
The numerical eigenvectors are orthonormal to the precision of the computation:
Diagonalization of the matrix r:
The diagonal elements are essentially the same as the eigenvalues:
The first eigenvector of a random matrix:
The position of the largest component in v:
Compute the eigenvalue corresponding to eigenvector v:
The general symbolic case quickly gets very complicated:
The expression sizes increase faster than exponentially:
Construct a 10,000×10,000 sparse matrix:
The eigenvector matrix is a dense matrix, and too large to represent:
Computing the few eigenvectors corresponding to the largest eigenvalues is much easier:
When eigenvalues are closely grouped, the iterative method for sparse matrices may not converge:
The iteration has not converged well after 1000 iterations:
You can give the algorithm a shift near an expected eigenvalue to speed up convergence:
The first four eigenvectors of a banded matrix:
A plot of the first four eigenvectors: