scipy.linalg.pinv#

scipy.linalg.pinv(a, *, atol=None, rtol=None, return_rank=False, check_finite=True)[source]#

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.
atolfloat, optional: Absolute threshold term, default value is 0.

Added in version 1.7.0.
rtolfloat, optional: Relative threshold term, default value is max(M, N) * eps where eps is the machine precision value of the datatype of a.

Added in version 1.7.0.
return_rankbool, optional: If True, return the effective rank of the matrix.
check_finitebool, 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.
rankint: 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] (1,2,3)

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:

ABA = A (B is a generalized inverse of A),
BAB = B (A is a generalized inverse of B),
(AB)* = AB (AB is hermitian),
(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