Haynsworth inertia additivity formula

Summary

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:

GeneralizationEdit

If H11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse H11 + instead of H11 −1.

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   and  .

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

See alsoEdit

Notes and referencesEdit

  1. ^ Haynsworth, E. V., "Determination of the inertia of a partitioned Hermitian matrix", Linear Algebra and its Applications, volume 1 (1968), pages 73–81
  2. ^ Zhang, Fuzhen (2005). The Schur Complement and Its Applications. Springer. p. 15. ISBN 0-387-24271-6.
  3. ^ The Schur Complement and Its Applications, p. 15, at Google Books
  4. ^ 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.