scipy.linalg.qz

scipy.linalg.qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False, overwrite_b=False)

QZ decompostion for generalized eigenvalues of a pair of matrices.

The QZ, or generalized Schur, decomposition for a pair of N x N nonsymmetric matrices (A,B) is:

(A,B) = (Q*AA*Z', Q*BB*Z')

where AA, BB is in generalized Schur form if BB is upper-triangular with non-negative diagonal and AA is upper-triangular, or for real QZ decomposition (output='real') block upper triangular with 1x1 and 2x2 blocks. In this case, the 1x1 blocks correspond to real generalized eigenvalues and 2x2 blocks are ‘standardized’ by making the corresponding elements of BB have the form:

[ a 0 ]
[ 0 b ]

and the pair of corresponding 2x2 blocks in AA and BB will have a complex conjugate pair of generalized eigenvalues. If (output='complex') or A and B are complex matrices, Z’ denotes the conjugate-transpose of Z. Q and Z are unitary matrices.

Parameters :

A : array_like, shape (N,N)

2-D array to decompose.

B : array_like, shape (N,N)

2-D array to decompose.

output : {‘real’,’complex’}, optional

Construct the real or complex QZ decomposition for real matrices. Default is ‘real’.

lwork : int, optional

Work array size. If None or -1, it is automatically computed.

sort : {None, callable, ‘lhp’, ‘rhp’, ‘iuc’, ‘ouc’}, optional

NOTE: THIS INPUT IS DISABLED FOR NOW, IT DOESN’T WORK WELL ON WINDOWS.

Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given a eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). For real matrix pairs, the sort function takes three real arguments (alphar, alphai, beta). The eigenvalue x = (alphar + alphai*1j)/beta. For complex matrix pairs or output=’complex’, the sort function takes two complex arguments (alpha, beta). The eigenvalue x = (alpha/beta). Alternatively, string parameters may be used:

• ‘lhp’ Left-hand plane (x.real < 0.0)
• ‘rhp’ Right-hand plane (x.real > 0.0)
• ‘iuc’ Inside the unit circle (x*x.conjugate() <= 1.0)
• ‘ouc’ Outside the unit circle (x*x.conjugate() > 1.0)

Defaults to None (no sorting).

Returns :

AA : ndarray, shape (N,N)

Generalized Schur form of A.

BB : ndarray, shape (N,N)

Generalized Schur form of B.

Q : ndarray, shape (N,N)

The left Schur vectors.

Z : ndarray, shape (N,N)

The right Schur vectors.

sdim : int, optional

If sorting was requested, a fifth return value will contain the number of eigenvalues for which the sort condition was True.

Notes

Q is transposed versus the equivalent function in Matlab.

New in version 0.11.0.

Examples

>>> from scipy import linalg
>>> np.random.seed(1234)
>>> A = np.arange(9).reshape((3, 3))
>>> B = np.random.randn(3, 3)

>>> AA, BB, Q, Z = linalg.qz(A, B)
>>> AA
array([[-13.40928183,  -4.62471562,   1.09215523],
       [  0.        ,   0.        ,   1.22805978],
       [  0.        ,   0.        ,   0.31973817]])
>>> BB
array([[ 0.33362547, -1.37393632,  0.02179805],
       [ 0.        ,  1.68144922,  0.74683866],
       [ 0.        ,  0.        ,  0.9258294 ]])
>>> Q
array([[ 0.14134727, -0.97562773,  0.16784365],
       [ 0.49835904, -0.07636948, -0.86360059],
       [ 0.85537081,  0.20571399,  0.47541828]])
>>> Z
array([[-0.24900855, -0.51772687,  0.81850696],
       [-0.79813178,  0.58842606,  0.12938478],
       [-0.54861681, -0.6210585 , -0.55973739]])