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][3]

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

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

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

See also

• Block matrix pseudoinverse

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