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The question is similar to this one Implicit derivative?

Let $x_1, x_2, x_3$ be three points in $\mathbb{R}^3$, $A=(a_{ij})$ is a $3\times 3$ matrix with $a_{ii}=0$ and $a_{ij}=\frac{1}{|x_i-x_j|}$ for $1\le i,j\le 3$. My question: what is $\frac{\partial A}{\partial x_1}$, $\frac{\partial A}{\partial x_2}$ and $\frac{\partial A}{\partial x_3}$? I came across this in a paper <'Note on an inequality' by Y.Xu 2006. Annales de l'Institut Henri Poincare>, but I was puzzled with this expressions.

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You don't think they're just the respective matrices of partial derivatives of each entry? –  Cam McLeman Apr 26 '10 at 4:09
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You probably should elaborate what you found puzzling... Cam's reading of the notation surely does not deserve that adjective, so you probably understood something else and/or the paper meant something else, but it is impossible to tell from what you wrote! –  Mariano Suárez-Alvarez Apr 26 '10 at 4:21
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I am puzzled by these partial derivatives, too. How do I take the partial derivative with respect to x_i of a_ij = 1/|x_i-x_j|, if x_i is a three-dimensional vector? I can't even do 1/|x_i|^2 . –  TonyK Apr 26 '10 at 9:06
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Some people use the partial derivative notation when they mean a partial differential. My guess would be that the author of the paper in question belongs to that category. –  fedja Apr 26 '10 at 14:57
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@miwalin: Please edit your question to include the citation to the paper. That will vastly improve it. –  Pete L. Clark Apr 26 '10 at 17:12
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1 Answer

I didn't look at the paper in too much detail, but my best guess is that $\partial_{x_i}A$ is a rank-3 tensor, in the sense that we can write $$v\cdot\partial_{x_i}A = \lim_{h\to 0} \frac{1}{h}(A[x_1, \ldots,x_i + hv, \ldots,x_p] - A[x_1,\ldots, x_p])$$

Or, in index notation, writing $a_{ij} = \frac{1}{x_i - x_j}$, we have that $$(\partial_{x_l}A)_{ijk} = \partial_{(x_l)^k} (\sum_{m} [(x_i)^m - (x_j)^m]^2)^{-1/2}$$ where $(x_l)^k$ denotes the $k$-th component of the point $x_l$. So in conjecture 1 of the linked paper, the term inside the absolute value sign in the first supremum is in fact, a vector, with the $i$ and $j$-th component in the above expression contracted against a vector $u$.

More intuitively, you can think of it as the gradient on the total space projected to the $k$-th copy of $\mathbb{R}^3$.

It is, admittedly, somewhat questionable and sloppy notation. I may have written it as the variation $\delta A / \delta x_i$ instead to get over the cognitive disconnect of taking a partial relative to a vector, but still it implicitly depends on the Euclidean structure of $\mathbb{R}^p$.

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I should probably add that rank-3 is not the best word for it. Let $V = \mathbb{R}^n, W = \mathbb{R}^m$, then $A: \oplus^m V \to W\otimes W$ is some matrix valued function. Then the $\partial_{x_i}A : \oplus^m V \to W\otimes W\otimes V^*$ is the tensor. –  Willie Wong Apr 26 '10 at 16:26
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