Stieltjes matrix resolvent ($G_X(z)=(X-zI{)}^{-1}$)

For a Hermitian matrix $X$ and nonreal $z$, this is the negative of the usual resolvent of an operator convention $(zI-X{)}^{-1}$. Its normalized trace is the Stieltjes transform of a measure of the empirical spectral measure with kernel $(x-z{)}^{-1}$. Naming the convention prevents sign errors in self-consistency identities.

For a zero-diagonal real symmetric matrix, put $q_i=x_i^T(X^{(i)}-zI{)}^{-1}{x}_i$ and $g_N=N^{-1}\mathrm{Tr}(X-zI{)}^{-1}$. The Schur complement formula for a diagonal resolvent entry gives $g_N=-N^{-1}{\sum }_i(z+q_i{)}^{-1}$. Subtracting the comparison value $-(z+g_N{)}^{-1}$ gives ${\epsilon }_N=N^{-1}{\sum }_i(q_i-g_N)/[(z+g_N)(z+q_i)]$. Upper-half-plane positivity and the principal minor resolvent trace bound control this defect by the average of $|q_i-g_N^{(i)}|$ plus a trace correction.

The eigenvalue interlacing of a Hermitian matrix and a principal minor bounds the difference of their resolvent traces by a constant times $|\mathrm{Im}z{|}^{-1}$. When both traces are normalized by the original dimension $N$, the bound acquires $N^{-1}$. One proof writes the difference using the interlacing counting functions, whose difference is at most one, and bounds the integral of $|t-z{|}^{-2}$ by $\pi /|\mathrm{Im}z|$. This gives the admissible absolute constant $c=\pi$.

By an orthonormal eigenbasis, the imaginary part of the Stieltjes matrix resolvent has eigenvalues $\eta /|\lambda -z{|}^2>0$. Thus its normalized trace has positive imaginary part, and its quadratic form at any vector has nonnegative imaginary part. This yields $|z+g|\ge \eta$ and $|z+q|\ge \eta$ for the normalized trace $g$ and a quadratic form $q$ used in Schur-complement estimates.