TheAlgorithms/Python · error · ValueError

Input matrix A is not invertible. Cannot compute Schur compl

Error message

Input matrix A is not invertible. Cannot compute Schur complement.

What it means

Raised by schur_complement when np.linalg.inv(mat_a) raises LinAlgError, i.e. block A is square but singular (determinant 0 or numerically rank-deficient). The mathematical Schur complement requires a non-singular A; the function offers the pseudo_inv parameter exactly for this case.

Source

Thrown at linear_algebra/src/schur_complement.py:54

        msg = (
            "Expected the same number of rows for A and B. "
            f"Instead found A of size {shape_a} and B of size {shape_b}"
        )
        raise ValueError(msg)

    if shape_b[1] != shape_c[1]:
        msg = (
            "Expected the same number of columns for B and C. "
            f"Instead found B of size {shape_b} and C of size {shape_c}"
        )
        raise ValueError(msg)

    a_inv = pseudo_inv
    if a_inv is None:
        try:
            a_inv = np.linalg.inv(mat_a)
        except np.linalg.LinAlgError:
            raise ValueError(
                "Input matrix A is not invertible. Cannot compute Schur complement."
            )

    return mat_c - mat_b.T @ a_inv @ mat_b


class TestSchurComplement(unittest.TestCase):
    def test_schur_complement(self) -> None:
        a = np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]])
        b = np.array([[0, 3], [3, 0], [2, 3]])
        c = np.array([[2, 1], [6, 3]])

        s = schur_complement(a, b, c)

        input_matrix = np.block([[a, b], [b.T, c]])

        det_x = np.linalg.det(input_matrix)
        det_a = np.linalg.det(a)

View on GitHub (pinned to f5988cc097)

Solutions

1. Supply a pseudo-inverse explicitly: schur_complement(a, b, c, pseudo_inv=np.linalg.pinv(a)).
2. Regularize A before calling, e.g. a_reg = a + 1e-8 * np.eye(a.shape[0]).
3. Inspect A's rank with np.linalg.matrix_rank(a) and remove linearly dependent rows/columns if full rank is expected.

Example fix

# before
result = schur_complement(a, b, c)  # a is singular

# after
result = schur_complement(a, b, c, pseudo_inv=np.linalg.pinv(a))

if np.linalg.matrix_rank(mat_a) < mat_a.shape[0]:
    result = schur_complement(mat_a, mat_b, mat_c, pseudo_inv=np.linalg.pinv(mat_a))
else:
    result = schur_complement(mat_a, mat_b, mat_c)

def is_invertible(a: np.ndarray) -> bool:
    return a.ndim == 2 and a.shape[0] == a.shape[1] and np.linalg.matrix_rank(a) == a.shape[0]

try:
    result = schur_complement(a, b, c)
except ValueError as e:
    if "not invertible" in str(e):
        result = schur_complement(a, b, c, pseudo_inv=np.linalg.pinv(a))
    else:
        raise

When it happens

Trigger: Passing a singular A such as np.array([[1, 2], [2, 4]]) without the pseudo_inv argument. Also happens for nearly-singular A under floating-point round-off when the LU solver reports exact singularity.

Common situations: Covariance matrices from degenerate data (fewer samples than dimensions), A containing linearly dependent rows/columns, or regularizing later but forgetting that this call needs the inverse.

Related errors

• No LU decomposition exists
• Matrix is not invertible
• Expected the same number of rows for A and B. Instead found
• Expected the same number of columns for B and C. Instead fou
• Coefficient matrix dimensions must be nxn but received {rows

AI-assisted analysis of TheAlgorithms/Python@f5988cc097 (2026-08-14). Data as JSON: /api/errors/6dd8bba1a185f833. Report an issue: GitHub.