I'm trying to prove one of these inequalities. This isn't a homework problem but trying to solve out of curiosity as it didn't have any relationship between $x$ and $\alpha$.
How do you prove: $$\alpha * ||x||^2 \leq ||Ax||^2 \leq \beta*||x||^2$$ where $$0 \leq \alpha\leq \beta$$ A is a matrix and $x$ is a real vector.
You cannot prove the statement with just what you have, as it is simply not true. If $A=I$ (the identity matrix) and $\alpha = 2$, then it is clear that if $x\neq 0$, $$\alpha ||x||^2 =2||x||^2>||x||^2=||Ax||^2$$