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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.

O.S
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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, $cov(M)=E[\{vec(M)-vec(E(M))\}\{vec(M)-vec(E(M))\}^T]$

Rizky Reza Fujisaki
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  • Yes, I also found it somewhere, but i don't know why they do it in this where. The formula used in (b) is it correct or not? – O.S Aug 24 '16 at 00:10
  • Is it a theorem or what? Please I need further details. (1) In your case the dimension of its covariance matrix is $n^2$x$n^2$ ? but I was thinking that the covariance matrix of an $n$x$n$ random matrix should be $n$x$n$. (2)What is its (i,j) entry in that case. (3) The formula used in (b) is it correct or not? – O.S Aug 24 '16 at 00:24
  • the formula in (b) is right for random vector, I do not know it is right for random matrix or not, since as I know, the definition of the cov for random matrix is as I wrote before – Rizky Reza Fujisaki Aug 24 '16 at 00:47
  • Why we have to use $vec$ for the $cov(M)$, because we have to exhaust all pairs of the entries (the random variables in the matrix), for comparation,if $V\in\mathbb{R}^2$ is a random vector with entries $X$ and $Y$, we use $cov(V)=E[{V-E(V)}{V-E(V)}^T]$ because by doing that, we can find $var(X)$, $var(Y)$, and $cov(X,Y)$. For our case, (let $M\in\mathbb{R}^{2\times 2}$ with entries $W$, $X$, $Y$, $Z$) by using $vec(M)$ to definne $cov(M)$, we will have all pairs as well, similar to the case $cov(V)$. If you use your definition in (b), you will lose many information (you can try it easily) – Rizky Reza Fujisaki Aug 24 '16 at 01:01