I have what is probably an elementary matrix calculus question.
Let $P$ be a linear operator and $||\cdot||$ be the ordinary unsquared vector norm.
Suppose one wants to minimize $||Px||$ subject to the trace of $P$ being a prime $p$, and $I \succeq P \succeq 0.$ How can one do this using Lagrangian multipliers? We know that the derivative of $||Px||$ is $(Px)/||Px||$, but how can we capture the other conditions?