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Proving Frobenius norm

  1. Sep 11, 2012 #1
    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. jcsd
  3. Sep 11, 2012 #2
    Do you know any vectorspaces with similar norms?
    Maybe you can relate the properties of those norms to this one!
  4. Sep 11, 2012 #3

    Ray Vickson

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

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