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 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 smaller (in modulus) than |
---|---|
Returns: |
B : (…, N, M) ndarray The pseudo-inverse of |
Raises: |
LinAlgError If the SVD computation does not converge. |
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). [R8787]
[R8787] | (1, 2) G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pp. 139-142. |
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
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https://docs.scipy.org/doc/numpy-1.14.2/reference/generated/numpy.linalg.pinv.html