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 H₁₁ is nonsingular and H₁₂^* is the conjugate transpose of H₁₂. The formula states:

where H/H₁₁ is the Schur complement of H₁₁ in H:

Generalization

If H₁₁ is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse instead of .

The formula does not hold if H₁₁ is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham, to the effect that and .

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

See also

• Block matrix pseudoinverse
• Sylvester's law of inertia

Notes and references