Haynsworth inertia additivity formula - Alchetron, the free social encyclopedia

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 ) )

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 ]

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 )

where H/H₁₁ is the Schur complement of H₁₁ in H:

H / H 11 = H 22 − H 12 ∗ H 11 − 1 H 12 .

References

Haynsworth inertia additivity formula Wikipedia

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