Symmetric positive semi-definite matrices and norm inequalities

Question

If $A$ and $B$ are symmetric positive semi-definite matrices such that $A \succeq B$ (i.e., $A-B$ is p.s.d.), does the following hold: $\Vert A\Vert_2 \geq \Vert B \Vert_2$?

Here $\Vert \cdot \Vert_2$ is the induced/operator 2-norm defined as

$\| A \|_2 = \sup \limits _{x \ne 0} \frac{\| A x\| _2}{\|x\|_2}$

Answer 1

Assuming $A$ and $B$ are symmetric:

Let $u$ be an unit eigen vector to eigen value $\|B\|_2$ of $B$. Then, we have $$ 0 \le u^T(A-B) u = u^T A u - u^T B u \le \|A\|_2 - \|B\|_2. $$