ordqz#
- scipy.linalg.ordqz(A, B, sort='lhp', output='real', overwrite_a=False, overwrite_b=False, check_finite=True)[source]#
- QZ decomposition for a pair of matrices with reordering. - The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details. - Parameters:
- A(N, N) array_like
- 2-D array to decompose 
- B(N, N) array_like
- 2-D array to decompose 
- sort{callable, ‘lhp’, ‘rhp’, ‘iuc’, ‘ouc’}, optional
- Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given an ordered pair - (alpha, beta)representing the eigenvalue- x = (alpha/beta), returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). For the real matrix pairs- betais real while- alphacan be complex, and for complex matrix pairs both- alphaand- betacan be complex. The callable must be able to accept a NumPy array. 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) 
 - With the predefined sorting functions, an infinite eigenvalue (i.e., - alpha != 0and- beta = 0) is considered to lie in neither the left-hand nor the right-hand plane, but it is considered to lie outside the unit circle. For the eigenvalue- (alpha, beta) = (0, 0), the predefined sorting functions all return False.
- outputstr {‘real’,’complex’}, optional
- Construct the real or complex QZ decomposition for real matrices. Default is ‘real’. 
- overwrite_abool, optional
- If True, the contents of A are overwritten. 
- overwrite_bbool, optional
- If True, the contents of B are overwritten. 
- check_finitebool, optional
- If true checks the elements of A and B are finite numbers. If false does no checking and passes matrix through to underlying algorithm. 
 
- Returns:
- AA(N, N) ndarray
- Generalized Schur form of A. 
- BB(N, N) ndarray
- Generalized Schur form of B. 
- alpha(N,) ndarray
- alpha = alphar + alphai * 1j. See notes. 
- beta(N,) ndarray
- See notes. 
- Q(N, N) ndarray
- The left Schur vectors. 
- Z(N, N) ndarray
- The right Schur vectors. 
 
 - See also - Notes - On exit, - (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues.- ALPHAR(j) + ALPHAI(j)*iand- BETA(j),j=1,...,Nare the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real generalized Schur form of (A,B) were further reduced to triangular form using complex unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is real; if positive, then the- jth and- (j+1)st eigenvalues are a complex conjugate pair, with- ALPHAI(j+1)negative.- Added in version 0.17.0. - Examples - >>> import numpy as np >>> from scipy.linalg import ordqz >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) >>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]]) >>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp') - Since we have sorted for left half plane eigenvalues, negatives come first - >>> (alpha/beta).real < 0 array([ True, True, False, False], dtype=bool)