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Module « scipy.linalg »

Fonction pinv - module scipy.linalg

Signature de la fonction pinv

def pinv(a, *, atol=None, rtol=None, return_rank=False, check_finite=True) 

Description

help(scipy.linalg.pinv)

Compute the (Moore-Penrose) pseudo-inverse of a matrix.

Calculate a generalized inverse of a matrix using its
singular-value decomposition ``U @ S @ V`` in the economy mode and picking
up only the columns/rows that are associated with significant singular
values.

If ``s`` is the maximum singular value of ``a``, then the
significance cut-off value is determined by ``atol + rtol * s``. Any
singular value below this value is assumed insignificant.

Parameters
----------
a : (M, N) array_like
    Matrix to be pseudo-inverted.
atol : float, optional
    Absolute threshold term, default value is 0.

    .. versionadded:: 1.7.0

rtol : float, optional
    Relative threshold term, default value is ``max(M, N) * eps`` where
    ``eps`` is the machine precision value of the datatype of ``a``.

    .. versionadded:: 1.7.0

return_rank : bool, optional
    If True, return the effective rank of the matrix.
check_finite : bool, optional
    Whether to check that the input matrix contains only finite numbers.
    Disabling may give a performance gain, but may result in problems
    (crashes, non-termination) if the inputs do contain infinities or NaNs.

Returns
-------
B : (N, M) ndarray
    The pseudo-inverse of matrix `a`.
rank : int
    The effective rank of the matrix. Returned if `return_rank` is True.

Raises
------
LinAlgError
    If SVD computation does not converge.

See Also
--------
pinvh : Moore-Penrose pseudoinverse of a hermitian matrix.

Notes
-----
If ``A`` is invertible then the Moore-Penrose pseudoinverse is exactly
the inverse of ``A`` [1]_. If ``A`` is not invertible then the
Moore-Penrose pseudoinverse computes the ``x`` solution to ``Ax = b`` such
that ``||Ax - b||`` is minimized [1]_.

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

Examples
--------

Given an ``m x n`` matrix ``A`` and an ``n x m`` matrix ``B`` the four
Moore-Penrose conditions are:

1. ``ABA = A`` (``B`` is a generalized inverse of ``A``),
2. ``BAB = B`` (``A`` is a generalized inverse of ``B``),
3. ``(AB)* = AB`` (``AB`` is hermitian),
4. ``(BA)* = BA`` (``BA`` is hermitian) [1]_.

Here, ``A*`` denotes the conjugate transpose. The Moore-Penrose
pseudoinverse is a unique ``B`` that satisfies all four of these
conditions and exists for any ``A``. Note that, unlike the standard
matrix inverse, ``A`` does not have to be a square matrix or have
linearly independent columns/rows.

As an example, we can calculate the Moore-Penrose pseudoinverse of a
random non-square matrix and verify it satisfies the four conditions.

>>> import numpy as np
>>> from scipy import linalg
>>> rng = np.random.default_rng()
>>> A = rng.standard_normal((9, 6))
>>> B = linalg.pinv(A)
>>> np.allclose(A @ B @ A, A)  # Condition 1
True
>>> np.allclose(B @ A @ B, B)  # Condition 2
True
>>> np.allclose((A @ B).conj().T, A @ B)  # Condition 3
True
>>> np.allclose((B @ A).conj().T, B @ A)  # Condition 4
True

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