Function pinv

numpy/linalg/_linalg.py:2154–2265

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 mat

(a, rcond=None, hermitian=False, *, rtol=_NoValue)

@array_function_dispatch(_pinv_dispatcher)
def pinv(a, rcond=None, hermitian=False, *, rtol=_NoValue):
    """
    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.

        .. versionadded:: 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 :math:`A^+`, is
    defined as: "the matrix that 'solves' [the least-squares problem]
    :math:`Ax = b`," i.e., if :math:`\\bar{x}` is said solution, then
    :math:`A^+` is that matrix such that :math:`\\bar{x} = A^+b`.

    It can be shown that if :math:`Q_1 \\Sigma Q_2^T = A` is the singular
    value decomposition of A, then
    :math:`A^+ = Q_2 \\Sigma^+ Q_1^T`, where :math:`Q_{1,2}` are
    orthogonal matrices, :math:`\\Sigma` is a diagonal matrix consisting
    of A's so-called singular values, (followed, typically, by