numpy.linalg.pinv¶

numpy.linalg.
pinv
(a, rcond=1e15)[source]¶ Compute the (MoorePenrose) pseudoinverse of a matrix.
Calculate the generalized inverse of a matrix using its singularvalue 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 pseudoinverted.
 rcond : (…) array_like of float
Cutoff for small singular values. Singular values smaller (in modulus) than rcond * largest_singular_value (again, in modulus) are set to zero. Broadcasts against the stack of matrices
Returns:  B : (…, N, M) ndarray
The pseudoinverse of a. If a is a matrix instance, then so is B.
Raises:  LinAlgError
If the SVD computation does not converge.
Notes
The pseudoinverse of a matrix A, denoted , is defined as: “the matrix that ‘solves’ [the leastsquares 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 socalled 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] (1, 2) G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pp. 139142. Examples
The following example checks that
a * a+ * a == a
anda+ * 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