Is there any general result characterizing real matrices $A$ such that $$[\mathrm{tr}(A)]^2\leq n\mathrm{tr}(A^2)?$$
I can see that the inequality holds if:
all eigenvalues of $A$ are real (by the Cauchy-Schwarz inequality) or
$A$ is a nonnegative matrix. To see this write $$n\mathrm{tr}(A^2)=n\sum_{i=1}^{n}(A_{ii})^{2}+n\sum_{i,j=1,i\neq j}^{n}A_{ij}A_{ji},$$ and note that, by the sum of squares inequality, $$n\sum_{i=1}^{n}(A_{ii})^{2}\geq\left( \sum_{i=1}^{n}A_{ii}\right)^{2}=\left[\mathrm{tr}(A)\right]^{2}.$$ If $A$ is nonnegative
$$n\sum_{i,j=1,i\neq j}^{n}A_{ij}A_{ji}\geq 0,$$ and therefore the inequality holds.
But what about matrices not satisfying 1. or 2.? Are there more general conditions (or other specific ones) under which the inequality above holds?