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Haynsworth inertia additivity formula

## Summary

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.[1]

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 nonsingular and H12* is the conjugate transpose of H12. The formula states:[2][3]

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

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

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

## Generalization

If H11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse H11 + instead of H11 −1.

The formula does not hold if H11 is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham,[4] to the effect that ${\displaystyle \pi (H)\geq \pi (H_{11})+\pi (H/H_{11})}$  and ${\displaystyle \nu (H)\geq \nu (H_{11})+\nu (H/H_{11})}$ .

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