QZ factorization for generalized eigenvalues
[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 |
| Produces a real decomposition with a quasitriangular |
If AA is triangular, then the diagonal elements a
= diag(AA) and b = diag(BB) are the generalized
eigenvalues that satisfy
A*V*b = B*V*a b'*W'*A = a'*W'*B
The eigenvalues produced by lambda = eig(A,B) are
the ratios of the diagonal elements a and b,
such that lambda = a./b.
If AA is not triangular, it is necessary
to further reduce the 2-by-2 blocks to obtain the eigenvalues of the
full system.