For real square matrices $A, B$ of compatible dimensions, does the following inequality hold?

Question

For real square matrices $A$ and $B$ of same dimension, does this hold?

$x^{T}A^{T}B^{T}BAx\leq\|A\|^2x^{T}B^{T}Bx$ for a non-zero vector $x$?

I know that $x^{T}A^{T}B^{T}BAx\leq\|A\|^2\|B\|^2\|x\|^2$ but I am not sure if the above inequality holds when $A$ and $B$ do not commute.

Any hints are greatly appreciated.

Answer 1

This is not true.

Counter Example: $$A = \begin{bmatrix} 2 & 7\\ 2 & 3 \end{bmatrix}$$

$$B = \begin{bmatrix} 2 & 3\\ 3 & 0 \end{bmatrix}$$ $$x = \begin{bmatrix} 2 \\ 7 \end{bmatrix}$$