Invariant Subspaces and Condition Numbers

The Schur form depends on the order of the eigenvalues on the diagonal of T and this may optionally be chosen by the user. Suppose the user chooses that $\lambda_1 , \ldots , \lambda_j$, $1 \leq j \leq n$, appear in the upper left corner of T. Then the first j columns of Z span the right invariant subspace of A corresponding to $\lambda_1 , \ldots , \lambda_j$.

The following routines perform this re-ordering and also compute condition numbers for eigenvalues, eigenvectors, and invariant subspaces:

1.

xTREXC will move an eigenvalue (or 2-by-2 block) on the diagonal of the Schur form from its original position to any other position. It may be used to choose the order in which eigenvalues appear in the Schur form.

2.

xTRSYL solves the Sylvester matrix equation $AX \pm XB=C$ for X, given matrices A, B and C, with A and B (quasi) triangular. It is used in the routines xTRSNA and xTRSEN, but it is also of independent interest.

3.

xTRSNA computes the condition numbers of the eigenvalues and/or right eigenvectors of a matrix T in Schur form. These are the same as the condition numbers of the eigenvalues and right eigenvectors of the original matrix A from which T is derived. The user may compute these condition numbers for all eigenvalue/eigenvector pairs, or for any selected subset. For more details, see section 4.8 and [12].

4.

xTRSEN moves a selected subset of the eigenvalues of a matrix T in Schur form to the upper left corner of T, and optionally computes the condition numbers of their average value and of their right invariant subspace. These are the same as the condition numbers of the average eigenvalue and right invariant subspace of the original matrix A from which T is derived. For more details, see section 4.8 and [12]

See Table 2.11 for a complete list of the routines.

Table 2.11: Computational routines for the nonsymmetric eigenproblem

Type of matrix	Operation	Single precision		Double precision
and storage scheme		real	complex	real	complex
general	Hessenberg reduction	SGEHRD	CGEHRD	DGEHRD	ZGEHRD
	balancing	SGEBAL	CGEBAL	DGEBAL	ZGEBAL
	backtransforming	SGEBAK	CGEBAK	DGEBAK	ZGEBAK
orthogonal/unitary	generate matrix after	SORGHR	CUNGHR	DORGHR	ZUNGHR
	Hessenberg reduction
	multiply matrix after	SORMHR	CUNMHR	DORMHR	ZUNMHR
	Hessenberg reduction
Hessenberg	Schur factorization	SHSEQR	CHSEQR	DHSEQR	ZHSEQR
	eigenvectors by	SHSEIN	CHSEIN	DHSEIN	ZHSEIN
	inverse iteration
(quasi)triangular	eigenvectors	STREVC	CTREVC	DTREVC	ZTREVC
	reordering Schur	STREXC	CTREXC	DTREXC	ZTREXC
	factorization
	Sylvester equation	STRSYL	CTRSYL	DTRSYL	ZTRSYL
	condition numbers of	STRSNA	CTRSNA	DTRSNA	ZTRSNA
	eigenvalues/vectors
	condition numbers of	STRSEN	CTRSEN	DTRSEN	ZTRSEN
	eigenvalue cluster/
	invariant subspace