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### 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 , , appear in the upper left corner of T. Then the first j columns of Z span the right invariant subspace of A corresponding to .

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 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 |  |  |  |

Next: Singular Value Decomposition Up: Nonsymmetric Eigenproblems Previous: Balancing Contents Index