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

\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}.

Generalization

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

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 $\pi (H)\geq \pi (H_{11})+\pi (H/H_{11})$ and $\nu (H)\geq \nu (H_{11})+\nu (H/H_{11})$ .

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

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
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.

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[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.

Haynsworth inertia additivity formula

Generalization

See also

Notes and references