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

# Eigenvalues

 Eigenvalues[m]gives a list of the eigenvalues of the square matrix m. Eigenvaluesgives the generalized eigenvalues of m with respect to a. Eigenvaluesgives the first k eigenvalues of m. Eigenvaluesgives the first k generalized eigenvalues.
• Eigenvalues finds numerical eigenvalues if m contains approximate real or complex numbers.
• Repeated eigenvalues appear with their appropriate multiplicity.
• An × matrix gives a list of exactly eigenvalues, not necessarily distinct.
• If they are numeric, eigenvalues are sorted in order of decreasing absolute value.
• The eigenvalues of a matrix are those for which for some non-zero eigenvector .
• The generalized eigenvalues of with respect to are those for which .
• When matrices m and a have a dimension- shared null space, then of their generalized eigenvalues will be Indeterminate.
• Ordinary eigenvalues are always finite; generalized eigenvalues can be infinite.
• For numeric eigenvalues, Eigenvalues gives the k that are largest in absolute value.
• Eigenvalues gives the k that are smallest in absolute value.
• The option settings Cubics->True and Quartics->True can be used to specify that explicit radicals should be generated for all cubics and quartics.
Exact eigenvalues:
Find approximate numerical eigenvalues:
Find eigenvalues starting with 20-digit precision:
Largest 5 eigenvalues:
Multiple eigenvalues are listed multiple times:
 Out[1]=
 Out[2]=

Exact eigenvalues:
 Out[1]=

Find approximate numerical eigenvalues:
 Out[1]=

Find eigenvalues starting with 20-digit precision:
 Out[1]=

Largest 5 eigenvalues:
 Out[1]=

Multiple eigenvalues are listed multiple times:
 Out[1]=
The largest 5 eigenvalues:
 Options   (1)
Explicitly use the cubic formula to give the result in terms of radicals:
 Applications   (2)
Smallest eigenvalue of a Hilbert matrix:
Eigenvalues of a random matrix:
Characteristic polynomial:
The general symbolic case very quickly gets very complicated:
The expression sizes increase faster than exponentially:
Here is a 20×20 Hilbert matrix:
Compute the smallest eigenvalue exactly and give its numerical value:
Compute the smallest eigenvalue with machine-number arithmetic:
The smallest eigenvalue is not significant compared to the largest:
Using sufficient precision for the numerical computation: