Haynsworth inertia additivity formula

{{short description|Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix}}

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.Haynsworth, E. V., "Determination of the inertia of a partitioned Hermitian matrix", Linear Algebra and its Applications, volume 1 (1968), pages 73–81

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:{{cite book |title=The Schur Complement and Its Applications |url=https://archive.org/details/schurcomplementi00zhan_673 |url-access=limited |page=[https://archive.org/details/schurcomplementi00zhan_673/page/n26 15]|first=Fuzhen |last=Zhang |year=2005 |publisher=Springer| isbn=0-387-24271-6 }}{{Google books |id=Wjd8_AwjiIIC |page=15 |title=The Schur Complement and Its Applications }}

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

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,{{cite journal |first=D. |last=Carlson |first2=E. V. |last2=Haynsworth |first3=T. |last3=Markham |title=A generalization of the Schur complement by means of the Moore–Penrose inverse |journal=SIAM J. Appl. Math. |volume=16 |issue=1 |year=1974 |pages=169–175 |doi=10.1137/0126013 }} to the effect that $\pi(H) \ge \pi(H_{11}) + \pi(H/H_{11})$ and $\nu(H) \ge \nu(H_{11}) + \nu(H/H_{11})$ .

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

Notes and references

Category:Linear algebra

Category:Matrix theory

Category:Theorems in algebra

Haynsworth inertia additivity formula

Generalization

See also

Notes and references