I have a bit of a problem with the Extended Cauchy Schwarz Inequality, specifically at the line
"with equality if and only if b = cB^(-1)d for some constant c."
As such, I'm having problems understanding the following lemma, at a similarly worded line:
"with the maximum attained when x = cB^(-1)d for any constant c != 0."
How do I demonstrate that the equality holds/maximum is attained if b = cB^(-1)d ? Most resources I found cater to general linear algebra courses, so it is very hard for me to relate to this text (Johnson & Wichern, Applied Multivariate Statistical Analysis).
Presumably you remember $x’y=x\cdot y = \|x\|\|y\|\cos\theta$ from your multivariable calculus classes. So the dot product is maximized when $\cos\theta=1$, which means $x$ is a positive scalar multiple of $y$.
Following the proof and applying this, equality holds here if and only if $B^{1/2}b$ is a (positive) scalar multiple of $B^{-1/2}d$. But $B^{1/2}b=cB^{-1/2}d$ means $b=cB^{-1}d$.