Haynsworth inertia additivity formula

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.
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:
where "H"/"H"11 is the Schur complement of "H"11 in "H":
Generalization.
If "H"11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse instead of .
The formula does not hold if "H"11 is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham, to the effect that formula_5 and formula_6.
Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold.