1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Skew-symmetric matrices and subspaces

  1. Jul 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Let W1 be the set of all nxn skew-symmetric matrices with entries from a field F. Assume F is not characteristic 2 and let W2 be a subspace of Mnxn(F) consisting of all nxn symmetric matrices. Prove the direct sum of W1 and W2 is Mnxn(F).


    2. Relevant equations



    3. The attempt at a solution

    I'll just do this for n = 3 for ease of formatting.

    Assume A is in W1 and B is in W2.
    A = [itex]\left( \begin{array}{ccc}
    0 & b & c \\
    -b & 0 & d \\
    -c & -d & 0 \end{array} \right)[/itex]

    B = [itex]\left( \begin{array}{ccc}
    e & f & g \\
    f & e & h \\
    g & h & e \end{array} \right)[/itex]

    A+B = [itex]\left( \begin{array}{ccc}
    e & b+f & c+g \\
    f-b & e & d+h \\
    g-c & h-d & e \end{array} \right)[/itex]

    For elements ax,y = ay,x in A they must both equal 0. ∴ W1 [itex]\cap[/itex] W2 = {0}

    So I have one part of the direct sum proof but the W1+W2 = Mnxn(F) part isn't working for me, clearly the diagonal entries can't be all the same.

    I guess the first thing I'm wondering about is how to use the assumption that F isn't characteristic 2. I don't see how that assumption helps me at all.
     
  2. jcsd
  3. Jul 20, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    The usual way to split a matrix A into symmetric and skew-symmetric is that (A+A^T)/2 is symmetric and (A-A^T)/2 is skew-symmetric (where ^T means transpose) and their sum is A. You are going to hit a glitch in characteristic 2. Because then 2=1+1 is not invertible.
     
  4. Jul 21, 2012 #3
    Interesting I haven't seen those definitions before.
    OK so the characteristic 2 condition is more of an edge case issue than anything?
     
  5. Jul 21, 2012 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    It's not really a definition. It's a trick to define the projections onto the subspaces. And you could think of characteristic 2 as an edge case, I suppose. But the fundamental problem is that, e.g., the nonzero matrix [[1,1],[1,1]] is both symmetric AND skew-symmetric in characteristic 2. In fact, there is no difference between being symmetric and skew-symmetric. Because x=(-x).
     
    Last edited: Jul 21, 2012
  6. Jan 13, 2013 #5
    thank you so much dick! you enlightened me about my homework due tmr!!!!!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Skew-symmetric matrices and subspaces
Loading...