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What is it //called// when you are doing linear regression on the problem: $ Y = AX+BZ $ where you are given observations Y and X and are assuming Z is independent Gausssians? If you do max-Likelihood I think you end up minimizing something like $\| Y-AX\|_B = (Y-AX)^TC^{-1}(Y-AX)$ where $C = B^T B$.

But what is this called and are there any tricks like linear regression or is the standard method just to optimize over the variables with some conditions on the rank of $B$ I believe.

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One key word would be "seemingly unrelated regressions" or SUR. The dependence of the noise term leads to different estimates of the regression coefficients (your $A$): http://en.wikipedia.org/wiki/Seemingly_unrelated_regressions

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