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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 (http://orion.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf). 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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  • $\begingroup$ 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. $\endgroup$
    – Denis Serre
    Commented Aug 29, 2012 at 14:37
  • $\begingroup$ I clarified what $X$ and $A$ are, that might help. $\endgroup$
    – user7807
    Commented Aug 29, 2012 at 14:39
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    $\begingroup$ chain rule; i think the question is better suited to math.SE and will receive more attention there. $\endgroup$
    – Suvrit
    Commented Aug 29, 2012 at 15:26
  • $\begingroup$ @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... $\endgroup$
    – user7807
    Commented Aug 29, 2012 at 17:46
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    $\begingroup$ The question was answered on Math.se :math.stackexchange.com/questions/188456/… Thanks for the tip Suvrit! $\endgroup$
    – user7807
    Commented Aug 29, 2012 at 18:54

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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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