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# scipy.linalg.pinv

scipy.linalg.pinv(a, *, atol=None, rtol=None, return_rank=False, check_finite=True, cond=<object object>, rcond=<object object>)

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

  Calculate a generalized inverse of a matrix using its singular-value decomposition `U @ S @ V` in the economy mode and picking up only the columns/rows that are associated with significant singular values.

  If `s` is the maximum singular value of `a`, then the significance cut-off value is determined by `atol + rtol * s`. Any singular value below this value is assumed insignificant.

  Parameters:
    a(M, N) array_like

      Matrix to be pseudo-inverted.

    atol float, optional

      Absolute threshold term, default value is 0.

      New in version 1.7.0.

    rtol float, optional

      Relative threshold term, default value is `max(M, N) * eps` where `eps` is the machine precision value of the datatype of `a`.

      New in version 1.7.0.

    return_rank bool, optional

      If True, return the effective rank of the matrix.

    check_finite bool, optional

      Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

    cond, rcond float, optional

      In older versions, these values were meant to be used as `atol` with `rtol=0`. If both were given `rcond` overwrote `cond` and hence the code was not correct. Thus using these are strongly discouraged and the tolerances above are recommended instead. In fact, if provided, atol, rtol takes precedence over these keywords.

      Deprecated since version 1.7.0: Deprecated in favor of `rtol` and `atol` parameters above and will be removed in SciPy 1.14.0.

      Changed in version 1.3.0: Previously the default cutoff value was just `eps*f` where `f` was `1e3` for single precision and `1e6` for double precision.

  Returns:
    B(N, M) ndarray

      The pseudo-inverse of matrix a.

    rank int

      The effective rank of the matrix. Returned if return_rank is True.

  Raises:
    LinAlgError

      If SVD computation does not converge.

  See also

  `pinvh`

    Moore-Penrose pseudoinverse of a hermitian matrix.

  Notes

  If `A` is invertible then the Moore-Penrose pseudoinverse is exactly the inverse of `A` [1]. If `A` is not invertible then the Moore-Penrose pseudoinverse computes the `x` solution to `Ax = b` such that `||Ax - b||` is minimized [1].

  References

  [1] (1,2,3)

  Penrose, R. (1956). On best approximate solutions of linear matrix equations. Mathematical Proceedings of the Cambridge Philosophical Society, 52(1), 17-19. doi:10.1017/S0305004100030929

  Examples

  Given an `m x n` matrix `A` and an `n x m` matrix `B` the four Moore-Penrose conditions are:

  1. `ABA = A` (`B` is a generalized inverse of `A`),

  2. `BAB = B` (`A` is a generalized inverse of `B`),

  3. `(AB)* = AB` (`AB` is hermitian),

  4. `(BA)* = BA` (`BA` is hermitian) [1].

  Here, `A*` denotes the conjugate transpose. The Moore-Penrose pseudoinverse is a unique `B` that satisfies all four of these conditions and exists for any `A`. Note that, unlike the standard matrix inverse, `A` does not have to be a square matrix or have linearly independent columns/rows.

  As an example, we can calculate the Moore-Penrose pseudoinverse of a random non-square matrix and verify it satisfies the four conditions.

      >>> import numpy as np
      >>> from scipy import linalg
      >>> rng = np.random.default_rng()
      >>> A = rng.standard_normal((9, 6))
      >>> B = linalg.pinv(A)
      >>> np.allclose(A @ B @ A, A)  # Condition 1
      True
      >>> np.allclose(B @ A @ B, B)  # Condition 2
      True
      >>> np.allclose((A @ B).conj().T, A @ B)  # Condition 3
      True
      >>> np.allclose((B @ A).conj().T, B @ A)  # Condition 4
      True