# Proving Frobenius norm

1. Sep 11, 2012

### tinorina

1. The problem statement, all variables and given/known data
Prove that the Frobenius norm is indeed a matrix norm.

2. Relevant equations
The definition of the the Frobenius norm is as follows:
||A||_F = sqrt{Ʃ(i=1..m)Ʃ(j=1..n)|A_ij|^2}

3. The attempt at a solution
I know that in order to prove that the Frobenius norm is indeed a matrix norm, it must satisfy the 3 properties of matrix norm, which are as follows:
1. f(A) >= 0, for all A in ℝ^(mxn) (f(A)=0 iff A=0)
2. f(A+B) <= f(A)+f(B), for all A, B in ℝ^(mxn)
3. f(αA) = |α|f(A), for all α in ℝ, A in ℝ^(mxn)

However, I'm not exactly sure how to go about proving each of the properties. Can someone please give me some hints? Thanks!

2. Sep 11, 2012

### conquest

Do you know any vectorspaces with similar norms?
Maybe you can relate the properties of those norms to this one!

3. Sep 11, 2012

### Ray Vickson

So, what difficulties are you having proving property 1? Where is your problem proving property 3?

RGV