I am reading Jorge Nocedal and Sepher J. Wright's Numerical Optimization and stuck at an exercise 4.6 in chapter 4.
The Canchy-Schwarz inequality states that for any vector $u$ and $v$, we have $$|u^Tv|\leq(u^Tu)(v^Tv)$$ with equality only $u$ and $v$ are parallel. When $B$ is positive definite, use this inequality to show that $$||g||^4\leq(g^TBg)(g^TB^{-1}g)$$ with equality only if $g$, $Bg$ and $B^{-1}g$ are parallel.
$B$ is positive definite if and only if $\forall g\in\mathbb R^n-\{0\}$, we have $$g^TBg>0$$ I have tried to prove this in some special cases:
$B=A^TA$. It is easy to prove.
When $B$ is symmetrical, we have a matrix $P$ to make $B=P^TP$. This is the case one.
However I have no idea in general case. There are some obstacles:
How do we deal with $B^{-1}$?
We cannot write $g^TBg$ as a form of norm.
Any advise is helpful.
I'm pretty sure Nocedal and Wright assume their positive definite matrices are symmetric. Indeed, the statement is false without that assumption. Try $$ B = \pmatrix{1 & t\cr -t & 1\cr}$$