Let $A,B \in {M_n}(R)$ be real $n \times n$ matrices and let matrices $|A|$ and $|B|$ contain the absolute values of the elements of $A$ and $B$ respectively. Construct the complex matrices $C = A + j \cdot B$ and $D = |A| + j \cdot |B|$, where $j$ is the imaginary unit.

For a square complex matrix $M \in {M_n}(C)$ its spectral norm is given by

$||M|{|_2} = \mathop {\max }\limits_{z \ne 0} \frac{{||Mz|{|_2}}}{{||M|{|_2}}} = \sqrt {{\lambda _{\max }}({M^*}M)} = {\sigma _{\max }}(M)$

Through random generation in MATLAB of 100000 matrices of different orders I obtain that the following inequality holds

$||C|{|_2} \le ||D|{|_2}$.

I have tried to prove it but I cannot succeed.

Have you encountered this inequality somewhere or can you prove it?