`numpy.linalg:pinv`

source: /numpy/linalg/_linalg.py :2153

Signature

   def     pinv (    a    ,    rcond    =  None   ,    hermitian    =  False   ,  * ,     rtol    =  <no value>     )

Summary

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

Extended Summary

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.
hermitian : bool, 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.
versionadded 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] $A x = b$ ," i.e., if $\overset{x}{ˉ}$ is said solution, then $A^{+}$ is that matrix such that $\overset{x}{ˉ} = A^{+} b$ .

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

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)))
np.allclose(B, np.dot(B, np.dot(a, B)))

`numpy.linalg:pinv`

Signature

Summary

Extended Summary

Parameters

Returns

Raises

Notes

Examples

See also

Aliases

Referenced by

This package