1
$\begingroup$

Let $A$ be a matrix in $SL_2(\mathbb R)$. Define the trace norm to be

$\|A\| = \sqrt{\mathrm{tr}(A^* A)}. $

Is it true that this norm satisfies some kind of multiplicative property; for example:

$\|AB\| \leq \|A\|\cdot\|B\|.$

Can someone give me a brief reference where basic properties of this norm are stated and proved?

Thanks.

  • 0
    @Alex: Thank you very much! If you could add that as an answer, and if no other answer comes along, then I could accept it!2011-09-17

1 Answers 1

1

As I said in my comment, this is also (and perhaps more commonly) called the Frobenius norm. The argument in the PDF that alex and I mentioned, showing that $\|AB\}\le\|A\|\cdot\|B\|$, also shows that the Frobenius norm of an $m\times n$ matrix $A$ is simply the ordinary Euclidean $2$-norm of $A$ (given by the Pythagorean theorem) if you think of $A$ as a vector in $\mathbb{R}^{mn}$, say be reading it out by rows. Thus, the basic properties of the Frobenius norm follow immediately from those of the $2$-norm.