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I have two symmetric square matrices $A$ and $B$ which are known in advance. Using these, I need to evaluate the following expression for many, many vectors:

$F(v) = \frac{||Av||}{||Av||+||Bv||}$

Because these matrices are symmetric I know that there must be some space in which they are diagonal - i.e. each represents a rotated non-uniform scale that scales orthogonal directions independently. Given this, my intuition is that I should be able to reduce the expression above to a dot product:

$G(v) = v \bullet k$

...where k is a vector of constants precomputed by performing some processing on the values of $F$ produced from using some set of eigenvectors as inputs. So far algebraic manipulation is failing me. Is a reduction to an expression of this form possible? If not, to what degree can the expression be simplified?

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    I suppose $vA$ is actually $Av$? – 2017-01-05
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    My mistake, I am used to using row vectors by default. – 2017-01-05
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    The amount of simplification you're looking for can't be achieved. Notice that $F(\lambda v)=F(v)$ for all $\lambda \neq 0$, but if $F$ could be written as $F(v)=\langle v, k\rangle$, then $F(\lambda v)=\lambda \cdot F(v)$. – 2017-01-05
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    That's a fair point. Does the situation change at all if I guarantee that all $v$ are normalized? – 2017-01-05
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    No. $F$ is always non-negative, but if $G(v)=\langle v, k \rangle$, the $G(-v)=-G(v)$ so if $k\neq 0$ then $G$ attains both positive and negative values. – 2017-01-05

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