The Frobenius matrix norm is defined for $m \times n$ real matrices as $$ \Vert A \Vert_F = + \sqrt{\mbox{Trace}(A^T A)} \tag{1} $$
For matrices $A$ and $B$ of sizes $(m \times n)$ and $(n \times k)$ respectively, the product $A B$ is well-defined.
In some text-books, I have seen the following identity (without proof): $$ \Vert A B \Vert_F \leq \Vert A \Vert_F \Vert B \Vert_F \tag{2} $$
The identity (2) is very useful in calculations in Numerical Linear Algebra.
I like to know how this can be proved.
Does the proof involve some properties in Matrix Algebra involving the trace operator?
Your comments are welcome!
