Haynsworth inertia additivity formula

The inertia of a Hermitian matrix H is defined as the ordered triple

${\displaystyle \mathrm {In} (H)=\left(\pi (H),\nu (H),\delta (H)\right)}$

whose components are respectively the numbers of positive, negative, and zero eigenvalues of H. Haynsworth considered a partitioned Hermitian matrix

${\displaystyle H={\begin{bmatrix}H_{11}&H_{12}\\H_{12}^{\ast }&H_{22}\end{bmatrix}}}$

where H₁₁ is

${\displaystyle \mathrm {In} {\begin{bmatrix}H_{11}&H_{12}\\H_{12}^{\ast }&H_{22}\end{bmatrix}}=\mathrm {In} (H_{11})+\mathrm {In} (H/H_{11})}$