Haynsworth inertia additivity formula

In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partitioned.^[1]

The inertia of a Hermitian matrix H is defined as the ordered triple

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

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

${\displaystyle H={\begin{bmatrix}H_{11}&H_{12}\\H_{12}^{\ast }&H_{22}\end{bmatrix}}}$

where H₁₁ is nonsingular and H₁₂^* is the conjugate transpose of H₁₂. The formula states:^[2][3]

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

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

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

Generalization

If H₁₁ is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse ${\displaystyle H_{11}^{+}}$  instead of ${\displaystyle H_{11}^{-1}}$ .

The formula does not hold if H₁₁ is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham,^[4] to the effect that ${\displaystyle \pi (H)\geq \pi (H_{11})+\pi (H/H_{11})}$  and ${\displaystyle \nu (H)\geq \nu (H_{11})+\nu (H/H_{11})}$ .

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

See also

• Block matrix pseudoinverse
• Sylvester's law of inertia

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
4. Carlson, D.; Haynsworth, E. V.; Markham, T. (1974). "A generalization of the Schur complement by means of the Moore–Penrose inverse". SIAM J. Appl. Math. 16 (1): 169–175. doi:10.1137/0126013.