= Positive imaginary part of a Stieltjes matrix resolvent
{title2=$\operatorname{Im}G_X(z)=\eta G_X(z)^*G_X(z)\quad(\eta=\operatorname{Im}z>0)$}
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 $|z+g|\geq\eta$ and $|z+q|\geq\eta$ for the normalized trace $g$ and a quadratic form $q$ used in Schur-complement estimates.
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