In my book the Frobenius Norm of a Matrix A is defined as $\|A\|_F =(\sum\limits_{i,j=1}^n |a_{i,j}|^2)^\frac{1}{2}$. Later on in my book the author uses $\|A^k\|_F$ and $\|A^k\|_F ^\frac{1}{k}$ but doesn't specify what this means exactly.
My guess would be $\|A^k\|_F =(\sum\limits_{i,j=1}^n |a_{i,j}|^k)^\frac{1}{2}$ and $\|A^k\|_F ^\frac{1}{k}=(\sum\limits_{i,j=1}^n |a_{i,j}|^k)^\frac{1}{k}$ but I'm not entirely sure, so I hope someone here can lift my uncertainties.
Many thanks in advance.
No. $||A^k||_F$ is the Frobenius norm of the Matrix $A^k$