What is the relation between $\lVert{Hp}\rVert^2$ and $\lVert{H}\rVert$ when $Hp\ne0$ and $\lVert{p}\rVert = 1$?

Question

What is the relation between $\lVert{Hp}\rVert^2$ and $\lVert{H}\rVert$ when $Hp\ne0$ and $\lVert{p}\rVert = 1$?

Note that $H$ and $p$ are sampled value of Gaussian distributed random variables, in which $H$ is a vector with a size of $1$ by $N$ and $p$ is a vector with a size of $N$ by $1$?

I think $$\lVert{Hp}\rVert^2 = \lVert{H}\rVert.$$ However, I failed to prove the above equation. Is my thought wrong? Thanks for reading my question.

Answer 1

All you can state is $$\|Hp\|^2 \ \le\ \|H\|^2$$ Can you prove this one?