Haynsworth inertia additivity formula

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.[1]

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:[2][3]

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,[4] 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

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.