I want to know the best way of computing the covariance matrix of a random matrix $M$.
Assume I have a $p$ by $p$ random matrix $S=\frac{1}{n}\sum_{t=1}^nV_tV'_t$ where $V_t$ are i.i.d $p$-random vectors drown from $N(0,W)$. Define $$M=\sqrt{n}(S-W),\;\; where \;\;E(S)=W.$$ I need a hint to start. Do I have to evaluate (a) or (b) defined as follow:
(a) $E(M_{ij}M_{uv})=nE((S_{ij}-W_{ij})(S_{uv}-W_{uv}))$
(b) Since $E(M)=0$, so its covariance matrix should be $Var(M)=E(MM')$ which $(i^{th},j^{th})$ entry is $$E(\sum_{k=1}^pM_{ik}M_{jk})=n\sum_{k=1}^pE((S_{ik}-W_{ik})(S_{jk}-W_{jk}))$$ Since $$C=AB \;\iff\; C_{ij}=\sum_{k}A_{ik}B_{kj}$$ Before I move forward in the computation, could you please indicate me which of the above formula is correct. Thanks in advance.
I don't remember where, but I am sure I read that before that to define $cov(M)$ where $M$ is random matrix, we need an transformation $vec:\mathbb{R}^{n\times n}\rightarrow\mathbb{R}^{n^2}$, where if $A=[a_{ij}]$ (the entries), then $vec(A)=[a_{11},a_{21},...,a_{n1},a_{12},a_{22},...,a_{n2},...,a_{1n},a_{2n},...,a_{nn}]^T$. Hence, $cov(M)=E[\{vec(M)-vec(E(M))\}\{vec(M)-vec(E(M))\}^T]$