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The Moore-Penrose\npseudoinverse is a unique ``B`` that satisfies all four of these\nconditions and exists for any ``A``. Note that, unlike the standard\nmatrix inverse, ``A`` does not have to be a square matrix or have\nlinearly independent columns/rows.\n\nAs an example, we can calculate the Moore-Penrose pseudoinverse of a\nrandom non-square matrix and verify it satisfies the four conditions.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import linalg\nrng = np.random.default_rng()\nA = rng.standard_normal((9, 6))\nB = linalg.pinv(A)\nnp.allclose(A @ B @ A, A) # Condition 1\nnp.allclose(B @ A @ B, B) # Condition 2\nnp.allclose((A @ B).conj().T, A @ B) # Condition 3\nnp.allclose((B @ A).conj().T, B @ A) # Condition 4\n", "execution_status": "success", "out": "True\n" }, { "__type": "Text", "__tag": 4046, "value": "\nIf the input array has more than two dimensions, it is interpreted as a batch of\ntwo-dimensional slices:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "a = np.stack((np.zeros((3, 3)), np.eye(3)))\np, ranks = linalg.pinv(a, return_rank=True)\np.shape\nranks\n", "execution_status": "success", "out": "array([0, 3])\n" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "pinvh", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.linalg._basic:pinvh" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Moore-Penrose pseudoinverse of a hermitian matrix." } ] } ], "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": "atol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "rtol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "return_rank", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "False" }, { "__type": "SigParam", "__tag": 4030, "name": "check_finite", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "True" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "pinv" }, "references": [ ".. [1] Penrose, R. (1956). On best approximate solutions of linear matrix", " equations. Mathematical Proceedings of the Cambridge Philosophical", " Society, 52(1), 17-19. :doi:`10.1017/S0305004100030929`." ], "qa": "scipy.linalg._basic:pinv", "arbitrary": [], "local_refs": [ "B", "a", "atol", "check_finite", "rank", "return_rank", "rtol" ] }