1. Not finding help here? Sign up for a free 30min 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!

Proof regarding skew symmetric matrices

  1. Oct 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that if A is skew symmetric, then Ak is skew symmetric for any positive odd integer k.

    2. Relevant equations



    3. The attempt at a solution

    Wow, I have no idea how to prove this. I'm guessing there's going to be induction involved. I know that the base case of k = 1 is true, because A1 = A, where A is skew symmetric. But after that, I don't know what to do at all.

    Any suggestions on where to go from here?
     
  2. jcsd
  3. Oct 14, 2009 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Skew-symmetric means A^(T)=(-A). (T=transpose). Does that help? Can you use that to show A^3 is skew-symmetric?
     
  4. Oct 14, 2009 #3
    [tex]
    \begin{align*}
    A &= -A^T\\
    A^3 &= (-A^T)^3\\
    &= (-A^T)(-A^T)(-A^T)\\
    &= (A^2)^T(-A^T)\\
    &= (-AA^2)^T\\
    &= (-A^3)^T
    \end{align*}
    [/tex]

    so A3 is skew symmetric.

    Thanks again for your help Dick. Now, how can I go about proving this for the rest of the positive odd integers. Will induction work? I'm trying to think of a way to do it, but I'm stuck on the "odd" integers part.
     
  5. Oct 14, 2009 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Sure, induction will work. But that's probably overkill. (A^n)^T=(A^T)^n. Just factor out the (-1)^n.
     
  6. Oct 14, 2009 #5
    OH...haha thanks very much. I tend to over analyze sometimes...
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof regarding skew symmetric matrices
Loading...