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
- [math]\displaystyle{ \mathrm{In}(H) = \left( \pi(H), \nu(H), \delta(H) \right) }[/math]
whose components are respectively the numbers of positive, negative, and zero eigenvalues of H. Haynsworth considered a partitioned Hermitian matrix
- [math]\displaystyle{ H = \begin{bmatrix} H_{11} & H_{12} \\ H_{12}^\ast & H_{22} \end{bmatrix} }[/math]
where H11 is nonsingular and H12* is the conjugate transpose of H12. The formula states:[2][3]
- [math]\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}) }[/math]
where H/H11 is the Schur complement of H11 in H:
- [math]\displaystyle{ H/H_{11} = H_{22} - H_{12}^\ast H_{11}^{-1}H_{12}. }[/math]
Generalization
If H11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse [math]\displaystyle{ H_{11}^+ }[/math] instead of [math]\displaystyle{ H_{11}^{-1} }[/math].
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 [math]\displaystyle{ \pi(H) \ge \pi(H_{11}) + \pi(H/H_{11}) }[/math] and [math]\displaystyle{ \nu(H) \ge \nu(H_{11}) + \nu(H/H_{11}) }[/math].
Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold.
See also
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. https://archive.org/details/schurcomplementi00zhan_673.
- ↑ 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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