Critical Points of Vector Function

Question

Given a matrix $A$ that is $n \times d$ and a $n$-dimensional vector $w$, define the vector function $f$ as:

$f(v) = \frac{w^TAv}{||Av||^2}$ where $v$ is a $d$-dimensional vector.

How can I find all of the critical points of $f(v)$ most efficiently? It can be either analytically or numerically. I need both the values of $f$ and $v$ that make up the critical point.

Answer 1

Hint: $f= gohoF$ where $g(z)= \langle w , z \rangle$,$\; h(y)=\frac{z}{\|z\|^2}$, and $F(v)=Av$.

Answer 2

There is no critical point except if $f=0$. -We assume that $w^TA\not= 0$ and $v$ is s.t. $Av\not= 0$-.

Indeed, the derivative is

$Df_v:h\in\mathbb{R}^d\mapsto (w^TAh||Av||^2-2w^TAvv^TA^TAh)/||Av||^4$. The critical points satisfy $||Av||^2w^TA-2w^TAvv^TA^TA=0$. That implies (multiply -right side- by $v$)

$w^TAv=0$ and, consequently $||Av||^2w^TA=0$, a contradiction.