Timeline for Covariance of random vector multiplied with a random matrix

6 events

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Jun 23, 2014 at 22:37	comment	added	whuber♦		(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.
Jun 23, 2014 at 21:44	comment	added	ws6079		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?
Jun 23, 2014 at 12:26	comment	added	whuber♦		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 $A\otimes A^\prime$.
Jun 23, 2014 at 12:14	comment	added	ws6079		@whuber thanks a lot! Assuming I can find the second moments of $A$, how would I continue?
Jun 23, 2014 at 12:11	comment	added	whuber♦		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 $i,j,l,m$.
Jun 23, 2014 at 7:45	history	asked	ws6079	CC BY-SA 3.0