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Homework Help: 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

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    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

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    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!!!!!
     
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