Eigenvalues, Eigenvectors and Schur Factorization

Let A be a square n-by-n matrix. A scalar $\lambda$ is called an eigenvalue and a non-zero column vector v the corresponding right eigenvector if $Av = \lambda v$ . A nonzero column vector u satisfying $u^H A = \lambda u^H$ is called the left eigenvector. The first basic task of the routines described in this section is to compute, for a given matrix A, all n values of $\lambda$ and, if desired, their associated right eigenvectors v and/or left eigenvectors u.

A second basic task is to compute the Schur factorization of a matrix A. If A is complex, then its Schur factorization is A=ZTZ^H, where Z is unitary and T is upper triangular. If A is real, its Schur factorization is A=ZTZ^T, where Z is orthogonal. and T is upper quasi-triangular (1-by-1 and 2-by-2 blocks on its diagonal). The columns of Z are called the Schur vectors of A. The eigenvalues of A appear on the diagonal of T; complex conjugate eigenvalues of a real A correspond to 2-by-2 blocks on the diagonal of T.

These two basic tasks can be performed in the following stages:

1.

A general matrix A is reduced to upper Hessenberg form H which is zero below the first subdiagonal. The reduction may be written A=QHQ^T with Q orthogonal if A is real, or A=QHQ^H with Q unitary if A is complex. The reduction is performed by subroutine xGEHRD, which represents Q in a factored form, as described in section 5.4. The routine xORGHR (or in the complex case xUNGHR) is provided to form Q explicitly. The routine xORMHR (or in the complex case xUNMHR) is provided to multiply another matrix by Q without forming Q explicitly.

2.

The upper Hessenberg matrix H is reduced to Schur form T, giving the Schur factorization H=STS^T (for H real) or H=STS^H (for H complex). The matrix S (the Schur vectors of H) may optionally be computed as well. Alternatively S may be postmultiplied into the matrix Q determined in stage 1, to give the matrix Z = Q S, the Schur vectors of A. The eigenvalues are obtained from the diagonal of T. All this is done by subroutine xHSEQR.

3.

Given the eigenvalues, the eigenvectors may be computed in two different ways. xHSEIN performs inverse iteration on H to compute the eigenvectors of H; xORMHR can then be used to multiply the eigenvectors by the matrix Q in order to transform them to eigenvectors of A. xTREVC computes the eigenvectors of T, and optionally transforms them to those of H or A if the matrix S or Z is supplied. Both xHSEIN and xTREVC allow selected left and/or right eigenvectors to be computed.

Other subsidiary tasks may be performed before or after those just described.