| MATLAB Function Reference | |
[AA,BB,Q,Z] = qz(A,B)
[AA,BB,Q,Z,V,W] = qz(A,B)
qz(A,B,flag)
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:
Produces a possibly complex decomposition with a triangular AA. For compatibility with earlier versions, 'complex' is the default. | |
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,
and
,
are the generalized eigenvalues that satisfy
The eigenvalues produced by
are the ratios of the
s and
s.
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.
For full matrices A and B, qz uses the LAPACK routines listed in the following table.
A and B Real | A or B Complex | |
|---|---|---|
A and B double | DGGES, DTGEVC (if you request the fifth output V) | ZGGES, ZTGEVC (if you request the fifth output V) |
A or B single | SGGES, STGEVC (if you request the fifth output V) | CGGES, CTGEVC (if you request the fifth output V) |
[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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