Consider a real $n\times n$ matrix $X$. Suppose I would like to minimize the sum of the squares of its entries as a penalty term in some convex minimization. I can write the term using the Frobenius norm $\operatorname{Tr}(X^T X)$ and then take the gradient w.r.t. $X$.
What if instead I want to minimize the sum of the $4^\text{th}$ power of the matrix entries? Is there a way I can write the penalty term in a compact form? (such that I can easily take the gradient w.r.t. $X$)
Thanks a lot.
Fabio