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I found the identity

$$ \frac{\partial( \det (X^T A X ))}{\partial X} = 2\det(X^TAX)AX(X^TAX)^{-1} $$

On the matrix cookbook ( It is equation 47 on p. 8. Note that $X$ is an $n \times m$ matrix and $A$ is a symmetric $n \times n$ matrix.

I could not find the identity in their cited references...does anyone know of a textbook or paper that has this identity?

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I don't understand the formula. On the one hand, the left-hand side should be linear form in $n^2$ variables, the differential of a scalar function with respect to $X$. On the other hand, the right-hand side is just a scalar. Both may not be equal. – Denis Serre Aug 29 '12 at 14:37
I clarified what $X$ and $A$ are, that might help. – user7807 Aug 29 '12 at 14:39
chain rule; i think the question is better suited to math.SE and will receive more attention there. – Suvrit Aug 29 '12 at 15:26
@Suvrit, easier said than done about the chain rule...I have tried this before: I have the identity (from a textbook) $D_x det(x) = det(x)x^{-T}$ and the Frechet derivative of $X^TAX$ is $D_x (X^T A X)h = x^TAh + h^TAX$. I find it hard to put the two together, especially because the Frechet derivative defintion of $x --- > det(x)$ involve a n inner product of matrices... – user7807 Aug 29 '12 at 17:46
The question was answered on… Thanks for the tip Suvrit! – user7807 Aug 29 '12 at 18:54

Some books to look for such things:

A M Mathai: "Jacobians of matrix transformations and functions nof matrix argument"

Magnus, Neudecker: "Matrix Differential Calculus with Applications in Statistics and Econometrics"

Kollo, von Rosen: "Advanced Multivariate Statistics with Matrices"

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Thank you but a short proof was recently given on… – user7807 Aug 29 '12 at 18:55

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