The Moore-Penrose pseudoinverse

A⁺ is defined for every matrix — square or not, singular or not — and it is computed from the SVD:

A = np.random.randn(5, 3)
A_plus = np.linalg.pinv(A)
print(A.shape, A_plus.shape)     # (5, 3) (3, 5) — the shape flips

U, S_, Vt = np.linalg.svd(A, full_matrices=False)
np.allclose(A_plus, Vt.T @ np.diag(1/S_) @ U.T)   # True

Note the shape flip: a tall matrix’s pseudoinverse is wide.