spectral norm of 2 matrices compared to spectral norm of the difference matrix In one paper I read, there is a notion that for matrix A and B of the same size (B is the sparsified version of A through random sparsification), the difference between the spectral norm is no greater than the spectral norm of the diff matrix, i.e :
 $| ||A|| - ||B|| | \leq  || A - B ||$ where $||A|| = \sqrt{\lambda_{max}}$ (spectral norm). Can anyone help to explain this inequality ? any help is greatly appreciated.
Thank you.

It's just a consequence of the triangle inequality, which is true for every norm.
$$ \|A\| = \|B + (A-B)\| \le \|B\| + \|A-B\|$$
so
$$ \|A\| - \|B\| \le \|A-B\|$$
and interchanging $A$ and $B$,  $$\|B\| - \|A\| \le \|B - A\| = \|A - B\|$$
so
$$ \left| \|A\| - \|B\| \right| = \max(\|A\|-\|B\|, \|B\|-\|A\|)\le \|A - B \|$$
