The QZ decomposition, also called the generalized Schur decomposition, of two square complex matrices 
and 
consists of unitary matrices 
and 
such that
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
and 
are upper triangular matrices. If the matrix pencil 
is regular, meaning that its determinant
is not identically zero, its generalized eigenvalues
are represented by the diagonal pairs 
. A pair with 
gives 
, while 
gives an infinite generalized
eigenvalue. A singular matrix pencil can produce an
indeterminate pair 
.
For real matrices, a real QZ decomposition uses orthogonal matrices and an upper quasitriangular
with an upper
triangular matrix 
.
The diagonal-pair rule applies to the 1-by-1 blocks. The 2-by-2 blocks encode conjugate
pairs of nonreal generalized eigenvalues,
which cannot be read from individual diagonal ratios.
