# xorbits.numpy.linalg.pinv#

xorbits.numpy.linalg.pinv(a, rcond=1e-15, hermitian=False)#

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.

Changed in version 1.14(numpy.linalg): Can now operate on stacks of matrices

Parameters
• a ((..., M, N) array_like) – Matrix or stack of matrices to be pseudo-inverted.

• rcond ((...) array_like of float) – 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.

• 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.

New in version 1.17.0(numpy.linalg).

Returns

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

Return type

(…, N, M) ndarray

Raises

LinAlgError – If the SVD computation does not converge.

scipy.linalg.pinv

Similar function in SciPy.

scipy.linalg.pinvh

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

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+:

>>> a = np.random.randn(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


Warning

This method has not been implemented yet. Xorbits will try to execute it with numpy.linalg.

This docstring was copied from numpy.linalg.