Proving the Frobenius Norm as a Matrix Norm

[tinorina]

Homework Statement

Prove that the Frobenius norm is indeed a matrix norm.

Homework 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}

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!

 

[conquest]

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

 

[Ray Vickson]

Homework Statement

Prove that the Frobenius norm is indeed a matrix norm.

Homework 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}

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!

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

RGV