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# numpy.linalg.pinv

linalg.pinv(a, rcond=None, hermitian=False, *, rtol=<no value>)

  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.

  Parameters:
    a(…, M, N) array_like

      Matrix or stack of matrices to be pseudo-inverted.

    rcond(…) array_like of float, optional

      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. Default: `1e-15`.

    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.

    rtol(…) array_like of float, optional

      Same as rcond, but it’s an Array API compatible parameter name. Only rcond or rtol can be set at a time. If none of them are provided then NumPy’s `1e-15` default is used. If `rtol=None` is passed then the API standard default is used.

      Added in version 2.0.0.

  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.

  See also

  `scipy.linalg.pinv`

    Similar function in SciPy.

  `scipy.linalg.pinvh`

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

  Notes

  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]

  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+`:

      >>> import numpy as np
      >>> rng = np.random.default_rng()
      >>> a = rng.normal(size=(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