Directional spectral measure of a Hermitian matrix

For positive integers $d,N$, let $X\in {\mathbb{C}}^{d\times N}$ and let $T\in {\mathbb{C}}^{d\times d}$ be Hermitian positive semidefinite, meaning $T=T^{\ast }$ and $v^{\ast }Tv\ge 0$ for every $v\in {\mathbb{C}}^d$. Let $T^{1\mathrm{/}2}$ be its Hermitian positive semidefinite square root and set $A:=\frac{1}{N}{T}^{1\mathrm{/}2}X{X}^{\ast }{T}^{1\mathrm{/}2}\mathrm{.}$ For each distinct eigenvalue $\lambda$ of the Hermitian matrix $A$, let $P_{\lambda }$ be the orthogonal projection onto its eigenspace. The empirical spectral distribution of $A$, normalized by its dimension $d$, is ${\mu }_A:=\frac{1}{d}{\sum }_{\lambda \in \mathrm{spec}(A)}\mathrm{rank}(P_{\lambda }){\delta }_{\lambda },$ where ${\delta }_{\lambda }$ is the unit point mass at $\lambda$. For a deterministic unit vector $x\in {\mathbb{C}}^d$, its directional spectral measure is ${\mu }_A^x:={\sum }_{\lambda \in \mathrm{spec}(A)}\mathrm{\parallel }{P}_{\lambda }x{\mathrm{\parallel }}_2^2{\delta }_{\lambda }\mathrm{.}$ For a Borel set $B\subseteq \mathbb{R}$, set ${\mathbf{1}}_B(A):={\sum }_{\lambda \in B\cap \mathrm{spec}(A)}{P}_{\lambda }$; then ${\mu }_A^x(B)=x^{\ast }{\mathbf{1}}_B(A)x$. Both measures are supported on the nonnegative real axis.

Bai, Miao, and Pan, §1, printed pp. 7–8 (full paper PDF).

Bai and Silverstein, Chapter 3, §§3.1–3.3, printed pp. 39–58 (full book PDF).