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}}.\,

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.
↑ The Schur Complement and Its Applications, p. 15, at Google Books

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