Suppose that $Q$ is a symmetric positive definite real matrix. Consider the real-valued function $$ f_Q(A) = \|A\| + \mathrm{trace}\big(Q(A^TA - 2A)\big). $$ Is $f_Q$ always minimized over matrices $A$ at a symmetric matrix? Or are there matrices $Q$, positive definite, such that the minimizing $A$ is not symmetric?
Above, $\|\cdot\|$ is the operator norm of $A$, for instance, given by the largest singular value of $A$. Note that this is a well posed question: by completing the square, we see that $$ f_Q(A) = \Big\{\|A\| + \mathrm{trace}\Big(Q (I- A)^T(I-A)\Big)\Big\} - \mathrm{trace}(Q). $$ Hence, one can see that $f_Q(A)$ is naturally bounded below, for each fixed $Q$.