`scipy.linalg._basic:pinv`

source: /scipy/linalg/_basic.py:1471

Signature

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

Summary

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

Extended Summary

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.

The a array argument 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 linalg_batch for details.

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.

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].

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
np.allclose(B @ A @ B, B)  # Condition 2
np.allclose((A @ B).conj().T, A @ B)  # Condition 3
np.allclose((B @ A).conj().T, B @ A)  # Condition 4

✓

True

If the input array has more than two dimensions, it is interpreted as a batch of two-dimensional slices:

a = np.stack((np.zeros((3, 3)), np.eye(3)))
p, ranks = linalg.pinv(a, return_rank=True)
p.shape
ranks

✓

array([0, 3])

Aliases

scipy.linalg.pinv