Haynsworth inertia additivity formula

In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth[1] (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:[2][3]

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

See also

Notes and references

  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
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