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For a random vector $x$ multiplied by a non-random matrix $A$, $y=Ax$ the covariance matrix of $y$ is given by

$\Sigma_y = E[Ax (Ax)^T] = E[Ax x^T A^T] = A E[x x^T ]A^T = A \Sigma_x A^T$,

where $\Sigma_x$ is the covariance matrix of $x$. How can I calculate $\Sigma_y$ if $A$ is a random matrix (uncorrelated with $x$) and $E[A]$ is known? The approach above does not seem to be directly applicable.

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  • $\begingroup$ You haven't enough information about $A$. You need all the second moments of $A = (a_{ij})$; that is, you need all the $\mathbb{E}(a_{il}a_{jm})$ for all possible indexes $i,j,l,m$. $\endgroup$
    – whuber ♦
    Commented Jun 23, 2014 at 12:11
  • $\begingroup$ @whuber thanks a lot! Assuming I can find the second moments of $A$, how would I continue? $\endgroup$
    – ws6079
    Commented Jun 23, 2014 at 12:14
  • $\begingroup$ It depends on what would constitute a "calculation" for you. Many people would hope for some simple-looking matrix identity, but I don't think one exists in this case. The closest you might come would be in terms of expectations of $x x^\prime$ as a vector and the related tensor $A\otimes A^\prime$. $\endgroup$
    – whuber ♦
    Commented Jun 23, 2014 at 12:26
  • $\begingroup$ I was hoping to be able to predict the autocovariance of the process $x$ observed through $A$. Could you please clarify what you mean by your last statement? $\endgroup$
    – ws6079
    Commented Jun 23, 2014 at 21:44
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    $\begingroup$ (I was thinking of multivariate $x$.) Writing $A=(a_{ij})$ and $x=(x_{kl})$, compute $$Axx^\prime A^\prime =a_{ij}x_{jk}x_{lk}a_{ml}=(a_{ij}a_{ml})(x_{jk}x_{lk})$$ (sum over all repeated subscripts). Writing $\mathbb{B}=(b_{im, jl})=(a_{ij}a_{ml})$ and $(y_{jl})=(x_{jk}x_{lk})$ and considering $im$ and $jl$ each as single indexes, this is the matrix product $\mathbb{B}y$ and the expectations of $\mathbb{B}$ and $y$ can be taken separately. $\endgroup$
    – whuber ♦
    Commented Jun 23, 2014 at 22:37

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