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Block LDL congruence adds the Hermitian pivot inertias

Statement

Let $k \geq 1$ , $m \geq 0$ , and let $H = (\begin{array}{cc} D & B^{*} \\ B & C \end{array}) \in C^{(k + m) \times (k + m)}$ be Hermitian, where $D = D^{*} \in C^{k \times k}$ is nonsingular, $B \in C^{m \times k}$ , and $C = C^{*} \in C^{m \times m}$ . Define the Schur complement $S = C - B D^{- 1} B^{*}$ . Inertia triples are added coordinatewise, and the $0 \times 0$ matrix has inertia $(0, 0, 0)$ . Then

inertia (H) = inertia (D) + inertia (S),

where the triples are added coordinatewise.

Facts & Assumptions

Given: $H, D, B, C, S$ as in the statement.

[F1]

Nonsingular complex congruence preserves Hermitian inertia (Complex Hermitian inertia is invariant under nonsingular congruence).

[F2]

A Hermitian matrix has an orthonormal eigenbasis with real eigenvalues (Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely).

Proof

1.1givenalgebra

Set $L = (\begin{array}{cc} I & 0 \\ B D^{- 1} & I \end{array})$ . It is invertible, with inverse $(\begin{array}{cc} I & 0 \\ - B D^{- 1} & I \end{array})$ . Direct block multiplication gives $H = L (\begin{array}{cc} D & 0 \\ 0 & S \end{array}) L^{*}$ , so $H$ is congruent to $diag (D, S)$ .

2.1givenF1F2step 1.1algebra∎

Since $D = D^{*}$ and $D$ is invertible, $(D^{- 1})^{*} = (D^{*})^{- 1} = D^{- 1}$ . Therefore $(B D^{- 1} B^{*})^{*} = B D^{- 1} B^{*}$ , and $C = C^{*}$ gives $S^{*} = S$ . By [F1], $inertia (H) = inertia (diag (D, S))$ . If $m = 0$ , the trailing block is empty and its inertia is $(0, 0, 0)$ by the stated convention. If $m > 0$ , [F2] diagonalizes the Hermitian matrices $D$ and $S$ separately; concatenating their orthonormal eigenbases diagonalizes the block diagonal matrix, so its positive, negative and zero counts are the sums of the corresponding counts for $D$ and $S$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Nicholas J. Higham, Numerical Linear Algebra and Matrix Analysis, MIMS EPrint 2015.103, §5.6 (standard reference, not scraped)