Subadditivity of the positive index: $n_+(Q_1 + Q_2) \le n_+(Q_1) + n_+(Q_2)$

For a Hermitian matrix $Q$ over an RCLike field, let $n_+(Q)$ (posIndex) be the number of strictly positive eigenvalues — equivalently, by Sylvester's law of inertia, the maximal dimension of a subspace on which the form $x \mapsto x^{\mathsf H} Q x$ is positive definite.

Statement. For Hermitian $m \times m$ matrices $Q_1, Q_2$,

The proof runs through the subspace characterization: a subspace on which $Q_1 + Q_2$ is positive definite meets the intersection of suitable complements, splitting its dimension between contributions counted by $n_+(Q_1)$ and $n_+(Q_2)$. This is the subadditivity statement appearing in Section 3 of the paper, just after the inertia lemma (lem:inertia).

In the module Zeta23.LinAlg.Inertia it is consumed by the zero-side block estimates Zeta23.ZeroSide.ZeroBlockData.posIndex_blockA_le and posIndex_blockQ_le, which control the number of positive eigenvalues of the block matrices arising in the matrix-variational argument for Theorem A.