numpy.linalg.pinv¶

numpy.linalg.pinv(a,rcond=1e-15)[source]¶

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

Calculate the generalized inverse of a matrix using itssingular-value decomposition (SVD) and including alllarge singular values.

Parameters:	a : (M, N) array_like Matrix to be pseudo-inverted. rcond : float Cutoff for small singular values.Singular values smaller (in modulus) thanrcond * largest_singular_value (again, in modulus)are set to zero.
Returns:	B : (N, M) ndarray The pseudo-inverse ofa. Ifa is amatrix instance, then soisB.
Raises:	LinAlgError If the SVD computation does not converge.

Notes

The pseudo-inverse of a matrix A, denoted, isdefined 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 singularvalue decomposition of A, then, where areorthogonal matrices, is a diagonal matrix consistingof A’s so-called singular values, (followed, typically, byzeros), and then is simply the diagonal matrixconsisting of the reciprocals of A’s singular values(again, followed by zeros).[R47]

References

[R47]

(1,2) G. Strang,Linear Algebra and Its Applications, 2nd Ed., Orlando,FL, Academic Press, Inc., 1980, pp. 139-142.

Examples

The following example checks thata*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