Haynsworth inertia additivity formula

In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partitioned. The inertia of a Hermitian matrix H is defined as the ordered triple I n ( H ) = ( π ( H ) , ν ( H ) , δ ( H ) ) {\displaystyle \mathrm {In} (H)=\left(\pi (H),\nu (H),\delta (H)\right)} whose components are respectively the numbers of positive, negative, and zero eigenvalues of H. Haynsworth considered a partitioned Hermitian matrix H = [ H 11 H 12 H 12 ∗ H 22 ] {\displaystyle H={\begin{bmatrix}H_{11}&H_{12}\\H_{12}^{\ast }&H_{22}\end{bmatrix}}} where H11 is nonsingular and H12* is the conjugate transpose of H12.

Source: Wikipedia — Haynsworth inertia additivity formula (CC BY-SA 4.0)