MATLAB Function Reference

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)

See Also

eig, hess, qz, rsf2csf

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.

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