If the matrix norm of A is defined as $\|A\|=\sum_{i,j}|Aij|$ then how do I determine if the sum of the diagonal elements, i.e., the trace is a valid norm?
I am not really sure how to approach this problem. Would it have to do anything with the max elements of a particular row/column of the matrix?
No: $Tr \left ( \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \right ) = 0$ yet this matrix is not zero. Putting in absolute values doesn't help either, since we have matrices like $\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$.
That said, the Frobenius norm $\| A \|_F$ is $\sqrt{Tr(A^T A)}$ and is indeed a norm, albeit not an induced norm.