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