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I'm currently watching a lecture on Machine Learning at Stanford university. The lecturer defines $f$ to be a mapping from $\mathbb{R}^{m \times n}$ to $\mathbb{R}$. I understand this to be a function that maps a matrix to a scalar value, i.e. trace, determinant.

However, he then goes on to say that the derivative of such a function is defined by the derivative of $f$ with respect to each element which to me would be $\frac{df}{dA_{mn}} : \mathbb{R} \to \mathbb{R}$. That doesn't make sense : / it's derivative would therefore have a different mapping, which can't be true.

My explanation isn't fantastic so refer to this link http://www.youtube.com/watch?v=5u4G23_OohI#t=3363s (it'll take you straight to the relevant time)

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    @MatthewHaworth: Not that I have any problems with it but why do you ask your question here? As far as I remember there is a Q&A forum for such questions on ml-class.org.2011-11-25

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The derivative of a multivariate function is called a Jacobian.

This is matrix whose entries are the partial derivatives you mentioned: $\frac{\partial f}{\partial A_{mn}}$

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    I've realised my misunderstanding, of course$f$does map $\mathbb{R}^{m \times n}$ to $\mathbb{R}$ but then taking the derivative with respect to A maps it back out to $\mathbb{R}^{m \times n}$ again. It was as simple as that :P2011-10-26