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 H11 is nonsingular and H12* is the conjugate transpose of H12.
The formula states:[2][3] where H/H11 is the Schur complement of H11 in H: If H11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse
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
Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold.