2×2 block matrices and the Schur complement #
This file proves properties of 2×2 block matrices
[A B; C D] that relate to the Schur complement
D - C⬝A⁻¹⬝B.
Main results #
matrix.det_from_blocks₂₂: determinant of a block matrix in terms of the Schur complement.
matrix.det_one_add_mul_comm: the Weinstein–Aronszajn identity.
matrix.schur_complement_pos_semidef_iff: If a matrix
Ais positive definite, then
[A B; Bᴴ D]is postive semidefinite if and only if
D - Bᴴ A⁻¹ Bis postive semidefinite.
LDU decomposition of a block matrix with an invertible top-left corner, using the Schur complement.
LDU decomposition of a block matrix with an invertible bottom-right corner, using the Schur complement.
Determinant of a 2×2 block matrix, expanded around an invertible top left element in terms of the Schur complement.
Determinant of a 2×2 block matrix, expanded around an invertible bottom right element in terms of the Schur complement.
The Weinstein–Aronszajn identity. Note the
1 on the LHS is of shape m×m, while the
the RHS is of shape n×n.
A special case of the Matrix determinant lemma for when
A = I.
TODO: show this more generally.