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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

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*:

If *H*_{11} 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 *H*_{11} is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham,^{[4]} to the effect that and .

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

**^**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**^**Carlson, D.; Haynsworth, E. V.; Markham, T. (1974). "A generalization of the Schur complement by means of the Moore–Penrose inverse".*SIAM J. Appl. Math*.**16**(1): 169–175. doi:10.1137/0126013.