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

\( \mathrm{In}(H) = \left( \pi(H), \nu(H), \delta(H) \right) \, \)

whose components are respectively the numbers of positive, negative, and zero eigenvalues of H. Haynsworth considered a partitioned Hermitian matrix

\( H = \begin{bmatrix} H_{11} & H_{12} \\ H_{12}^\ast & H_{22} \end{bmatrix} \)

where H11 is nonsingular and H12* is the conjugate transpose of H12. The formula states:[2][1]

\( \mathrm{In} \begin{bmatrix} H_{11} & H_{12} \\ H_{12}^\ast & H_{22} \end{bmatrix} = \mathrm{In}(H_{11}) + \mathrm{In}(H/H_{11}) \)

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

\( H/H_{11} = H_{22} - H_{12}^\ast H_{11}^{-1}H_{12}. \, \)

Generalization

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,[3] to the effect that \( \pi(H) \ge \pi(H_{11}) + \pi(H/H_{11}) and \nu(H) \ge \nu(H_{11}) + \nu(H/H_{11}) \) .

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

Block matrix pseudoinverse

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

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