Theorem: Full-basis Hermitian certificate

Let $H=H^{†}$, $L=\mathrm{diag}({\mu }_1,\dots ,{\mu }_d)$, with sorted real ${\mu }_i$, and $A\in {\mathbb{C}}^{d\times d}$. Suppose validated bounds establish

Set

Then, counted with multiplicity,

No indexed level is omitted.

Proof. The polar normalization $Q=A(A^{†}A{)}^{-1/2}$ is unitary. Singular values give $\Vert A-Q\Vert \le a$, $\Vert A\Vert \le \sqrt{1+\delta }$, hence

The Hermitian min–max principle compares each ordered eigenvalue with that of $QL{Q}^{†}$, namely ${\mu }_i$.