Just a brief addition to Denis's answer:

The function that you have **is convex** for unitarily invariant norms, but for the (basis dependent) elementwise absolute value, it can clearly break. 

Here is a simple counterexample:

\begin{equation*}
 X = \begin{pmatrix}
 2 & 0 & 0\\\\
 0 & 1 & 1\\\\
 0 & 0 & 1
\end{pmatrix},\qquad Y = \begin{pmatrix}
 10  &    9 &    5\\\\
     9 &   10 &    5\\\\
     5 &    5 &    4
\end{pmatrix}
\end{equation*}

Now, simply define $Z = 0.5(X+Y)$. Then, we see that $f(Z) = 33$, while $0.5(f(X)+f(Y)) = 3$, clearly violating convexity.