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## 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!