Let us assume that $B$ is a symmetric positive semi-definite matrix in $\mathbb{R}^{m\times m}$. What is the optimum solution for the following constrained quadratic programming problem: $$min_{X>=0}\,\,trace(X^TBX).$$ Here, it should be mentioned that X is a non-negative matrix.
Moreover, if we assume that $X$ is a non-negative diagonal matrix, how can we solve this problem with this new constraint?
P.S.: In both cases, I guess that the solution is the zero matrix, which minimizes the objective function subject to the specified constraints.