What is the relationship between $||.||_{max}$ and energy of a matrix?

Question

I was interested to find a relationship between $||.||_{max}$, i.e. max-norm of a matrix with that of its energy. $||.||_{max}$ of a matrix is defined by the maximum entry in the matrix. Generally, energy of a matrix is defined by the sum of diagonal elements ($\sigma$) of the matrix after singular value decomposition. I found, an inequality concerning representation of energy of a non-negative matrix in terms of $||.||_{max}$. But, what happens when the constrained of non-negativity is lifted? Is there any generalized relationship?