I found the connection between Frobenius norm and trace operation is simple and elegant.
Frobenius norm is a square-root sum of absolute squares of every element. It defines as:
If all elements in the matrix is real number, then we do not need an absolute operator.
Now, we re-express matrix X such that each row is one vector:
Since the sum of square of is a dot product , we can compute the sum of squres of all elements by:
Any element along the diagonal of is a sum square of element from one of the row of
To compute the total sum squares of every element, we can sum along the diagonal:
And a sum along a diagonal of matrix is a trace operation.