# Definition of matrix derivative

Let $A$ be an $m \times n$ matrix. Let $a_{ij}$ be an element of $A$. What does the notation $\frac{\partial A}{\partial a_{ij}}$ mean?

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en.wikipedia.org/wiki/Matrix_calculus –  anon Oct 9 '12 at 19:20
In the absence of any clarification in the context, I'd say it was a complicated way to write $e_{ij}$, the matrix with $1$ at position $i,j$ and $0$ everywhere else. –  Henning Makholm Oct 9 '12 at 19:26
Suppose I have the matrix A = [x y; y x]. Would $\frac{\partial A}{\partial a_{11}}$ = [1 0; 0 1]? This seems like it is what wikipedia is saying, but paper I am reading says otherwise. It says $\frac{\partial A}{\partial a_{11}}$ = [1 0; 0 0] –  Stuart Oct 9 '12 at 19:26
@user19192: In that case I don't think $\frac{\partial A}{\partial a_{11}}$ has any clear meaning, since $a_{11}$ is not an independent variable that goes into the definition of $A$. Do you have more context of where you have seen this? –  Henning Makholm Oct 9 '12 at 19:29
Ok, I understand. A is just a function of its m*n inputs. Not a concrete matrix. –  Stuart Oct 9 '12 at 19:30
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