# Haynsworth inertia additivity formula

In mathematics, the **Haynsworth inertia additivity formula**, discovered by Emilie Virginia Haynsworth^{[1]} (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

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

where *H*_{11} is nonsingular and *H*_{12}^{*} is the conjugate transpose of *H*_{12}. The formula states:^{[2]}^{[3]}

where *H*/*H*_{11} is the Schur complement of *H*_{11} in *H*:

## See also

## Notes and references

- ↑ Haynsworth, E. V., "Determination of the inertia of a partitioned Hermitian matrix",
*Linear Algebra and its Applications*, volume 1 (1968), pages 73–81 - ↑ Zhang, Fuzhen (2005).
*The Schur Complement and Its Applications*. Springer. p. 15. ISBN 0-387-24271-6. - ↑
*The Schur Complement and Its Applications*, p. 15, at Google Books

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