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Suppose $X$ is an $m\times n$ random matrix where rows $x^T$ are I.I.D. samples of $n$-dimensional Gaussian, and $A,B,C$ are matrices of appropriate dimension. What is the value of the following matrix?

$$E[X^T A X B X^T CX]$$

Special cases of this equation occur in the literature for special settings of $A,B,C$, see earlier post here. I'm looking for the general case. Formula is likely to be more compact when parameterized in terms of $H=E[xx']$ and $\mu=E[x]$ rather than covariance $\Sigma$

Some examples of special cases.

From Seber's "Matrix Handbook for Statisticians":

  1. For Gaussian $x$ enter image description here

  2. For Wishart $W$ enter image description here

  3. For Gaussian $x$ we can let $A,B,C=I$, reshaping/summing the resulting matrix in various ways gives identities here

Yaroslav Bulatov
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  • I had been thinking about answering this question but just did not find the time and energy to do it befor the bounty. So now I was picking it up and I encounter something unsurprising. I am not sure I got this right: In your case if $m=1$ then the $x$ is a row vector, right? So that is different from your first example from Seber where the $x$ is a column? – Sextus Empiricus Oct 03 '22 at 06:34
  • yes, for $m=1$, $X$ is a row vector. BTW, I got a candidate answer on mathematica stack exchange for the formula, but need to find a way to simplify it – Yaroslav Bulatov Oct 04 '22 at 04:47
  • So the result is a matrix right? And not some scalar number like in the first example with a single Gaussian vector $x$. I feel like I am getting stuck already on a very simple matrix multiplication, but with $X^T$ being a column vector in the front, you never are gonna get a scalar as outcome. – Sextus Empiricus Oct 04 '22 at 06:59
  • Or did you mean to write $H = E[x'x]$ which will be a matrix as well if $x$ is a 1xn row vector. – Sextus Empiricus Oct 04 '22 at 07:05
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    The literature takes x to be a column vector, and data matrix X stacks them as rows. So X and x are transposes of each other when m=1. I did mean xx' which is Hessian. x'x corresponds to Gram matrix. Aka X'X is Hessian, XX' is gram – Yaroslav Bulatov Oct 11 '22 at 17:43
  • But you take $x$ to be a n-dimensional row vector right? Stacking them to obtain a m x n matrix. – Sextus Empiricus Oct 11 '22 at 17:47
  • What do you mean by 'stacks them as rows'? – Sextus Empiricus Oct 11 '22 at 17:50
  • No, x is a column vector. Corresponding data matrix X is a "row vector" consisting of x transposed – Yaroslav Bulatov Oct 11 '22 at 17:51
  • But if $x$ is a column vector how can you say "rows $x$ are I.I.D. samples of n-dimensional Gaussian" that makes $x$ a row vector. – Sextus Empiricus Oct 11 '22 at 17:54
  • Maybe missing a transpose there. In a data matrix , examples are rows. But a standalone example is a column vector, it's confusing I know – Yaroslav Bulatov Oct 11 '22 at 18:03
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    Currently I am interpreting your matrix $X$ as following. You have $m$ iid observations $a,b,c,etc.$ drawn from a $n$-dimensional Gaussian distribution and you stack them as rows. $$ X = \begin{bmatrix} a_1, a_2, \dots , a_n \ b_1, b_2, \dots , b_n \ c_1, c_2, \dots , c_n \ \vdots \ etc. \end{bmatrix}$$ – Sextus Empiricus Oct 11 '22 at 19:36

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