pinv(a, rtol=None, 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.


rtol param is requested in array-api-standard in instead of a parameter in official NumPy operator.

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

  • rtol ((..) {float or ndarray of float}, optional) – Cutoff for small singular values. Singular values less than or equal to rtol * 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.


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

Return type

(.., N, M) ndarray


MXNetError – If the SVD computation does not converge.


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



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(2, 3) >>> pinv_a = np.linalg.pinv(a) >>> (a -,, a))).sum() array(0.) >>> (pinv_a -,, pinv_a))).sum() array(0.)