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Let $A=\begin{pmatrix} 1 & 2 & 3 \\ 1 & 1 & 1 \end{pmatrix}$ and $\operatorname{SomeOperation}(A)=\begin{pmatrix}1*2*3 \\ 1*1*1\end{pmatrix} =\begin{pmatrix}6 \\ 1\end{pmatrix}$. What is the operation $\operatorname{SomeOperation}$?

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It is whatever you want to call it. Too unnatural to have a standard name. – anon Aug 25 '13 at 13:34
I'd call it muckification. – Gerry Myerson Aug 25 '13 at 13:58

If $A > 0$, you can express this using elementwise functions $$ exp(log(A)\cdot u) $$ Where $u$ is a vector of ones.

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Can you give your definition of the logarithm for a matrix? – hhh Aug 25 '13 at 18:47
@hhh: It's on Wikipedia. (And requires $A$ be square, which is inherent in the positive-definiteness condition Glynne writes, $A>0$.) – anon Aug 25 '13 at 20:55

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