Consider the PSD symmetric invertible $k\times k$ block matrix $$M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}, $$ where $A\in R^{k_1\times k_1}$, $B\in R^{k_1\times k_2}$ ($C=B^T$), $D\in R^{k_2\times k_2}$, $A,B$ are PSD and symmetric.
Let $x\neq 0$ is a $k$-dimensional vector. I would like to find bounds (lower/upper) for some norm of the quadratic form $W=x^T M x$ in terms of the blocks. Something like (this might not be the case): $$ ||W||\leq f(||A||,||B||,||C||,||D||,x), $$ where $f$ is some function of the blocks and vector $x$.
I have tried to use the Rayleigh-Ritz theorem, it gives bounds for the norm of $W$, but not considering the blocks, I was wondering if you could adapt it for the blocks structure of $M$. Thanks in advance.
Using Cauchy–Schwarz, the Frobenius norm, and $y=Mx,\;$ one obtains $$\eqalign{ x^Ty &\le \|x\|\cdot\|y\| \\&= \|x\|\cdot\|Mx\| \\ &\le \|x\|\cdot\|M\|\cdot\|x\| \\ &\le \|x\|^2\cdot\|M\| \\ &= \|x\|^2\;\Big(\|M\|^2\Big)^{1/2} \\ &= \|x\|^2\;\Big(\|A\|^2 + 2\|B\|^2 + \|D\|^2\Big)^{1/2} \\ }$$ The Frobenius norm is crucial for utilizing the block structure of $M$.