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. The inertia of a Hermitian matrix H is defined as the ordered triple
I n ( H ) = ( π ( H ) , ν ( H ) , δ ( H ) ) {\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
H = [ H 11 H 12 H 12 ∗ H 22 ] {\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:
I n [ H 11 H 12 H 12 ∗ H 22 ] = I n ( H 11 ) + I n ( H / H 11 ) {\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:
H / H 11 = H 22 − H 12 ∗ H 11 − 1 H 12 . {\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 H 11 + {\displaystyle H_{11}^{+}} instead of H 11 − 1 {\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, to the effect that π ( H ) ≥ π ( H 11 ) + π ( H / H 11 ) {\displaystyle \pi (H)\geq \pi (H_{11})+\pi (H/H_{11})} and ν ( H ) ≥ ν ( H 11 ) + ν ( H / H 11 ) {\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