#### Schur Complement

p
```import unittest

import numpy as np
import pytest

def schur_complement(
mat_a: np.ndarray,
mat_b: np.ndarray,
mat_c: np.ndarray,
pseudo_inv: np.ndarray | None = None,
) -> np.ndarray:
"""
Schur complement of a symmetric matrix X given as a 2x2 block matrix
consisting of matrices A, B and C.
Matrix A must be quadratic and non-singular.
In case A is singular, a pseudo-inverse may be provided using
the pseudo_inv argument.

Link to Wiki: https://en.wikipedia.org/wiki/Schur_complement
See also Convex Optimization - Boyd and Vandenberghe, A.5.5
>>> import numpy as np
>>> a = np.array([[1, 2], [2, 1]])
>>> b = np.array([[0, 3], [3, 0]])
>>> c = np.array([[2, 1], [6, 3]])
>>> schur_complement(a, b, c)
array([[ 5., -5.],
[ 0.,  6.]])
"""
shape_a = np.shape(mat_a)
shape_b = np.shape(mat_b)
shape_c = np.shape(mat_c)

if shape_a[0] != shape_b[0]:
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)
det_s = np.linalg.det(s)

assert np.is_close(det_x, det_a * det_s)

def test_improper_a_b_dimensions(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]])

with pytest.raises(ValueError):
schur_complement(a, b, c)

def test_improper_b_c_dimensions(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, 3], [6, 3, 5]])

with pytest.raises(ValueError):
schur_complement(a, b, c)

if __name__ == "__main__":
import doctest

doctest.testmod()
unittest.main()
```