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how would I calculate the derivative of the following. I want to know the derivative so that I can maximise it.

$$ \frac{x^TAx}{x^TBx} $$

Both the matricies A and B are symmetric. I know the derivative of $\frac{d}{dx}x^TAx = 2Ax$. Haven't been very successful applying the quotient rule to the above though. Appreciate the help. Thanks!

EDIT: In response to "What goes wrong when applying the chain rule". We know that: $$ \frac{d}{dx}\frac{u}{v} = \frac{vu' - uv'}{v^2} $$ Which would give me: $$ \frac{2x^TBxAx - 2x^TAxBx}{x^TBx^2} \, or \, \frac{2Axx^TBx - 2Bxx^TAx}{(x^TBx)^2} $$

In the first case the dimensions don't agree. In the second they do, but I don't want to assume that it's correct just because the dimensions agree. If it is correct then please do let me know!

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What goes wrong when you try to apply the chain rule? – Owen Biesel Oct 13 '12 at 4:47
See edit. Thanks – foges Oct 13 '12 at 4:57
up vote 3 down vote accepted

Both expressions are correct, up to some missing parentheses, in the following sense: For each $i$, we have $\frac{\partial}{\partial x_i}(x^TAx) = 2(Ax)_i$ and $\frac{\partial}{\partial x_i}(x^TBx) = 2(Bx)_i$, so the quotient rule tells us that

$$\frac{\partial}{\partial x_i}\frac{x^TAx}{x^TBx} = \frac{2(Ax)_i(x^TBx) - 2(Bx)_i(x^TAx)}{(x^TBx)^2}.$$

This may be written, if you interpret $d/dx$ as gradient, as

$$\frac{d}{dx}\frac{x^TAx}{x^TBx} = \frac{2Ax(x^TBx) - 2Bx(x^TAx)}{(x^TBx)^2} = \frac{2Axx^TBx - 2Bxx^TAx}{(x^TBx)^2},$$ so your second expression is correct.

Since $(x^TAx)$ and $(x^TBx)$ are just numbers, if left inside their parentheses they can multiply a vector or matrix from either side. Hence the first expression is also correct if you are sure to leave in the parentheses:

$$\frac{d}{dx}\frac{x^TAx}{x^TBx} = \frac{2(x^TBx)Ax - 2(x^TAx)Bx}{(x^TBx)^2}.$$

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Ok great, thanks. That makes sense :) – foges Oct 13 '12 at 5:27

Use hypoograph/Epigraph Technique to maximize/minimize the ratio. For example

$\min_x \frac{x^T A x}{x^T B x}$

$ \equiv \min_{x,t} t$

$ \text{ subject to }$

$ {x^T A x}\leq t(x^T B x),t>0$.

Then form a Lagrangain to solve for the optimum.

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