Let $A$ be a $p \times q$ real matrix of full column rank $q$, and let $u, v$ be two real vectors of (euclidean) norm 1.
I want to know whether the following inequality holds:
$$ v^TA^TAv \geq v^T A^T u u^T A v, $$ with equality if and only if $u$ ($v$) is the left (right) singular vector associated with any singular value of $A$.
As trivial as I think it must be, I can't wrap my head around this.
[Edit] I could come up with a solution using the Cauchy-Schwarz inequality. Letting $y= Av$, we have
$$ v^T A^T A v = y^T y = <y, y> <v, v> \quad\geq\quad <y, v>^2 = y^Tvv^Ty = v^TA^Tv v^TA v $$ So while I could not prove the "if and only if" part, it seems I could at least prove the inequality.