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


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

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


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

where H11 is nonsingular and H12* is the conjugate transpose of H12. The formula states:


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

where H/H11 is the Schur complement of H11 in H:


 * $$ H/H_{11} = H_{22} - H_{12}^\ast H_{11}^{-1}H_{12}. $$

Generalization
If H11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse $$H_{11}^+$$ instead of $$H_{11}^{-1}$$.

The formula does not hold if H11 is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham, to the effect that $$ \pi(H) \ge \pi(H_{11}) + \pi(H/H_{11}) $$ and $$ \nu(H) \ge \nu(H_{11}) + \nu(H/H_{11}) $$.

Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold.