### What is the Haynsworth Inertia Additivity Formula?

Start with a Hermitian matrix, $\bb{H}$ – which is one that’s equal to its conjugate transpose $\bb{H^*}$ ((That is to say: switch the rows and columns, then take the complex conjugate of every entry – if this gives you the original matrix, it’s Hermitian.)).

Haynsworth defined the inertia of a Hermitian matrix to be a triple, $\text{In}(\bb{H}) = \br{ \pi(\bb{H}), \nu(\bb{H}), \delta(\bb{H})}$, where:

• $\pi(\bb{H})$ is the number of positive eigenvalues of $\bb{H}$;
• $\nu(\bb{H})$ is the number of negative eigenvalues of $\bb{H}$;
• $\delta(\bb{H})$ is the number of zero eigenvalues of $\bb{H}$.

If you split a Hermitian matrix into parts, $\bb{H} = \mattwotwo{\bb H_{11}}{\bb H_{12}}{\bb H^*_{12}}{\bb H_{22}}$ (where $\bb{H_{11}}$ is non-singular) then there’s a relationship between the inertias of the parts:

$\text{In}(\bb{H}) = \text{In}\br{\bb{H_{11}}} + \text{In}\br{\bb{H}/ bb{H_{11}}}$

What’s the $/$ here? It’s not division, but a Schur complement – $\bb{H}/\bb{H_{11}} = \bb{H_{22}} - \bb{H^*_{12}} \bb{H^{-1}_{11}} \bb{H_{12}}$.

This means, if we’re just interested in the signs of the eigenvalues of a Hermitian matrix, we can break it into smaller parts and work with those instead.

### Who was Emilie Virginia Haynsworth

Haynsworth was born in Sumter, South Carolina, in 1916. After studying at Coker College and Columbia University in New York, she taught mathematics in high school, worked at the Aberdeen Proving Ground during WW2 and then lectured at the University of Illinois.

She completed her doctorate at UNC Chapel Hill in 1952, then worked at Wilson College in Pennsylvania, the National Bureau of Statistics, and finally at Auburn University. Her research into the eigenvalues of matrices was renowned for its originality and elegance.

Haynsworth retired in 1983 and died in South Carolina in 1985.