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numpy.linalg.pinv — NumPy v2.3 Manual

linalg.pinv(a, rcond=None, hermitian=False, *, rtol=<no value>)[source]#

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

Calculate the generalized inverse of a matrix using its singular-value decomposition (SVD) and including all large singular values.

Parameters:

a(…, M, N) array_like

Matrix or stack of matrices to be pseudo-inverted.

rcond(…) array_like of float, optional

Cutoff for small singular values. Singular values less than or equal to rcond * largest_singular_value are set to zero. Broadcasts against the stack of matrices. Default: 1e-15.

hermitianbool, optional

If True, a is assumed to be Hermitian (symmetric if real-valued), enabling a more efficient method for finding singular values. Defaults to False.

rtol(…) array_like of float, optional

Same as rcond, but it’s an Array API compatible parameter name. Only rcond or rtol can be set at a time. If none of them are provided then NumPy’s 1e-15 default is used. If rtol=None is passed then the API standard default is used.

New in version 2.0.0.

Returns:

B(…, N, M) ndarray

The pseudo-inverse of a. If a is a matrix instance, then so is B.

Raises:

LinAlgError

If the SVD computation does not converge.

Notes

The pseudo-inverse of a matrix A, denoted \(A^+\), is defined as: “the matrix that ‘solves’ [the least-squares problem] \(Ax = b\),” i.e., if \(\bar{x}\) is said solution, then \(A^+\) is that matrix such that \(\bar{x} = A^+b\).

It can be shown that if \(Q_1 \Sigma Q_2^T = A\) is the singular value decomposition of A, then \(A^+ = Q_2 \Sigma^+ Q_1^T\), where \(Q_{1,2}\) are orthogonal matrices, \(\Sigma\) is a diagonal matrix consisting of A’s so-called singular values, (followed, typically, by zeros), and then \(\Sigma^+\) is simply the diagonal matrix consisting of the reciprocals of A’s singular values (again, followed by zeros). [1]

References

[1]

G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pp. 139-142.

Examples

The following example checks that a * a+ * a == a and a+ * a * a+ == a+:

>>> import numpy as np
>>> rng = np.random.default_rng()
>>> a = rng.normal(size=(9, 6))
>>> B = np.linalg.pinv(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True

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