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schur - Schur decomposition

Syntax

T = schur(A) T = schur(A,flag) [U,T] = schur(A,...)

Description

The schur command computes the Schur form of a matrix.

T = schur(A) returns the Schur matrix T.

T = schur(A,flag) for real matrix A, returns a Schur matrix T in one of two forms depending on the value of flag:

`'complex'`	`T` is triangular and is complex if `A` has complex eigenvalues.
`'real'`	`T` has the real eigenvalues on the diagonal and the complex eigenvalues in 2-by-2 blocks on the diagonal. `'real'` is the default.

If A is complex, schur returns the complex Schur form in matrix T. The complex Schur form is upper triangular with the eigenvalues of A on the diagonal.

The function rsf2csf converts the real Schur form to the complex Schur form.

[U,T] = schur(A,...) also returns a unitary matrix U so that A = U*T*U' and U'*U = eye(size(A)).

Examples

H is a 3-by-3 eigenvalue test matrix:

H = [ -149    -50   -154
       537    180    546
       -27     -9    -25 ]

Its Schur form is

schur(H) 

ans =
     1.0000   -7.1119 -815.8706
          0    2.0000  -55.0236
          0         0    3.0000

The eigenvalues, which in this case are 1, 2, and 3, are on the diagonal. The fact that the off-diagonal elements are so large indicates that this matrix has poorly conditioned eigenvalues; small changes in the matrix elements produce relatively large changes in its eigenvalues.

Algorithm

Input of Type Double

If A has type double, schur uses the LAPACK routines listed in the following table to compute the Schur form of a matrix:

Matrix A	Routine
Real symmetric	`DSYTRD`, `DSTEQR` `DSYTRD`, `DORGTR`, `DSTEQR` (with output `U`)
Real nonsymmetric	`DGEHRD`, `DHSEQR` `DGEHRD`, `DORGHR`, `DHSEQR` (with output `U`)
Complex Hermitian	`ZHETRD`, `ZSTEQR` `ZHETRD`, `ZUNGTR`, `ZSTEQR` (with output `U`)
Non-Hermitian	`ZGEHRD`, `ZHSEQR` `ZGEHRD`, `ZUNGHR`, `ZHSEQR` (with output `U`)

Input of Type Single

If A has type single, schur uses the LAPACK routines listed in the following table to compute the Schur form of a matrix:

Matrix A	Routine
Real symmetric	`SSYTRD`, `SSTEQR` `SSYTRD`, `SORGTR`, `SSTEQR` (with output `U`)
Real nonsymmetric	`SGEHRD`, `SHSEQR` `SGEHRD`, `SORGHR`, `SHSEQR` (with output `U`)
Complex Hermitian	`CHETRD`, `CSTEQR` `CHETRD`, `CUNGTR`, `CSTEQR` (with output `U`)
Non-Hermitian	`CGEHRD`, `CHSEQR` `CGEHRD`, `CUNGHR`, `CHSEQR` (with output `U`)

References

[1] Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J.Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK User's Guide (http://www.netlib.org/lapack/lug/lapack_lug.html), Third Edition, SIAM, Philadelphia, 1999.