numpy.linalg.pinv

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: 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.

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 , is defined as: “the matrix that ‘solves’ [the least-squares problem] ,” i.e., if is said solution, then is that matrix such that .

It can be shown that if is the singular value decomposition of A, then , where are orthogonal matrices, is a diagonal matrix consisting of A’s so-called singular values, (followed, typically, by zeros), and then 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