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/H11 is the Schur complement of H11 in H:
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.
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.
- D. Carlson, E. V. Haynsworth, and T. Markham, "A generalization of the Schur complement by means of the Moore–Penrose inverse", SIAM J. Appl. Math., volume 16(1) (1974), pages 169–175