Eigenvalues and Eigensystems

Overview

This chapter describes the routines used to calculate the eigenvalues and the eigenvectors of a matrix. There are routines for real symmetric, real nonsymmetric and complex hermitian eigensystems. Eigenvalues can be computed with or without eigenvectors. The hermitian and real symmetric matrix algorithms are symmetric bidiagonalization followed by QR reduction. The nonsymmetric algorithm is the Francis QR double-shift.

Eigenvalues of a real matrix

eigs(m)
Return a real vector with the eigenvalues of the real symmetrix matrix m. It is left to the user to ensure that the matrix is symmetric.
eigsv(m)

Gives a real symmetrix matrix m returns a real vector with the eigenvalues and a real matrix with the corresponding eigenvectors arranged by columns. It is left to the user to ensure that the matrix is symmetric.

Example:

m = new(4, 4, |i,j| 1/(i+j-1))
e, v = eigsv(m)

-- the following expression will give a diagonal matrix with
-- the eigenvalues along the diagonal
print(prod(v,mul(m,v)))
eigh(m)
Return a complex vector with the eigenvalues of the complex hermitian matrix m. It is left to the user to ensure that the matrix is actually hermitian. The eigenvalues are unordered.
eighv(m)
Gives a complex hermitian matrix m returns a complex vector with the unordered eigenvalues and a complex matrix with the corresponding eigenvectors arranged by columns. It is left to the user to ensure that the matrix is actually hermitian.
eigns(m)
Return a complex vector with the eigenvalues of the real nonsymmetric matrix m.
eignsv(m)
Gives a real nonsymmetrix matrix m returns a complex vector with the eigenvalues and a complex matrix with the corresponding eigenvectors arranged by columns. Please note that, even if the original matrix is real the resulting eigenvalues and eigenvectors will be complex.
schur(m)

This function works like the function eignsv() but it does return the Schur form matrix T and the schur vector matrix Z. These matrices satisfies the identity:

T = Z^{-1} m Z

where m is the given real non-symmetrix matrix.

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