TITLE: Frobenius norm of a matrix
QUESTION [7 upvotes]: I know that Frobenius norm of a matrix A is equal to the square root of the trace of (A*conjugate transpose(A)). But how do I prove it mathematically?

REPLY [7 votes]: Presumably, you have the entry-wise definition,
$$
\|A\|_F^2 = \sum_{i,j} |a_{ij}|^2
$$
In order to get to this from the trace of the product $AA^*$, use the summation definition of matrix multiplication:
$$
[AB](i,j) = \sum_{k=1}^n a_{ik}b_{kj}
$$
and of the trace:
$$
\text{trace}(A) = \sum_{\ell = 1}^n a_{\ell \ell}
$$