qz (MATLAB Functions)

QZ factorization for generalized eigenvalues

Syntax

[AA,BB,Q,Z,] = qz(A,B)
[AA,BB,Q,Z,V,W] = qz(A,B)
qz(A,B,flag)

Description

The qz function gives access to intermediate results in the computation of generalized eigenvalues.

[AA,BB,Q,Z] = qz(A,B) for square matrices A and B, produces upper quasitriangular matrices AA and BB, and unitary matrices Q and Z such that Q*A*Z = AA, and Q*B*Z = BB. For complex matrices, AA and BB are triangular.

[AA,BB,Q,Z,V,W] = qz(A,B) also produces matrices V and W whose columns are generalized eigenvectors.

qz(A,B,flag) for real matrices A and B, produces one of two decompositions depending on the value of flag:

'complex'
Produces a possibly complex decomposition with a triangular AA. For compatibility with earlier versions, 'complex' is the default.

'real'
Produces a real decomposition with a quasitriangular AA, containing 1-by-1 and 2-by-2 blocks on its diagonal.

`'complex'`	Produces a possibly complex decomposition with a triangular `AA`. For compatibility with earlier versions, `'complex'` is the default.
`'real'`	Produces a real decomposition with a quasitriangular `AA`, containing 1-by-1 and 2-by-2 blocks on its diagonal.

If AA is triangular, the diagonal elements of AA and BB,

```
alpha = diag(AA)
beta = diag(BB)
```

are the generalized eigenvalues that satisfy

A*V*diag(beta) = B*V*diag(alpha)
diag(beta)*W'*A = diag(alpha)*W'*B

The eigenvalues produced by

```
lambda = eig(A,B)
```

are the element-wise ratios of alpha and beta.

```
lambda = alpha ./ beta
```

If AA is not triangular, it is necessary to further reduce the 2-by-2 blocks to obtain the eigenvalues of the full system.

Algorithm

For real QZ on real A and real B, eig uses the LAPACK DGGES routine. If you request the fifth output V, eig also uses DTGEVC.

For complex QZ on real or complex A and B, eig uses the LAPACK ZGGES routine. If you request the fifth output V, eig also uses ZTGEVC.

See Also

eig

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.

quiver3 rand