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The eigenvalue " }, { "__type": "InlineCode", "__tag": 4051, "value": "x = (alphar + alphai*1j)/beta" }, { "__type": "Text", "__tag": 4046, "value": ". For complex matrix pairs or output='complex', the sort function takes two complex arguments (alpha, beta). The eigenvalue " }, { "__type": "InlineCode", "__tag": 4051, "value": "x = (alpha/beta)" }, { "__type": "Text", "__tag": 4046, "value": ". 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If false does no checking and passes matrix through to underlying algorithm." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The QZ, or generalized Schur, decomposition for a pair of n-by-n matrices (A,B) is " } ] }, { "__type": "Code", "__tag": 4050, "value": "(A,B) = (Q @ AA @ Z*, Q @ BB @ Z*)", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "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 (" }, { "__type": "InlineCode", "__tag": 4051, "value": "output='real'" }, { "__type": "Text", "__tag": 4046, "value": ") 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 " } ] }, { "__type": "Code", "__tag": 4050, "value": "[ a 0 ]\n[ 0 b ]", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "and the pair of corresponding 2x2 blocks in AA and BB will have a complex conjugate pair of generalized eigenvalues. If (" }, { "__type": "InlineCode", "__tag": 4051, "value": "output='complex'" }, { "__type": "Text", "__tag": 4046, "value": ") or A and B are complex matrices, Z' denotes the conjugate-transpose of Z. Q and Z are unitary matrices." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "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 " }, { "__type": "InlineRole", "__tag": 4003, "value": "linalg_batch", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " for details. 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The QZ decomposition is not unique, so\ndepending on the underlying library that is used, there may be\ndifferences in the signs of coefficients in the following output.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "AA, BB, Q, Z = qz(A, B)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "AA\nBB\nQ\nZ\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nVerify the QZ decomposition. With real output, we only need the\ntranspose of ``Z`` in the following expressions.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "Q @ AA @ Z.T # Should be A\nQ @ BB @ Z.T # Should be B\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nRepeat the decomposition, but with ``output='complex'``.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "AA, BB, Q, Z = qz(A, B, output='complex')\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nFor conciseness in the output, we use ``np.set_printoptions()`` to set\nthe output precision of NumPy arrays to 3 and display tiny values as 0.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "np.set_printoptions(precision=3, suppress=True)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "AA\nBB\nQ\nZ\n", "execution_status": "failure" }, { "__type": "Text", "__tag": 4046, "value": "\nWith complex arrays, we must use ``Z.conj().T`` in the following\nexpressions to verify the decomposition.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "Q @ AA @ Z.conj().T # Should be A\nQ @ BB @ Z.conj().T # Should be B\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "ordqz", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.linalg._decomp_qz:ordqz" }, "kind": "module" }, "descriptions": [], "type": "func" } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "A", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "B", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "output", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "real" }, { "__type": "SigParam", "__tag": 4030, "name": "lwork", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "sort", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "overwrite_a", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "overwrite_b", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "check_finite", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "True" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "qz" }, "references": null, "qa": "scipy.linalg._decomp_qz:qz", "arbitrary": [], "local_refs": [ "A", "AA", "B", "BB", "Q", "Z", "check_finite", "lwork", "output", "overwrite_a", "overwrite_b", "sort" ] }