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!

Similar matrices please check my proof.

  1. Dec 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that every periodic matrix is similar to a diagonal matrix.

    2. Relevant equations

    I use ~ to denote A similar to B as A~B

    3. The attempt at a solution

    Let A be periodic, and let A^m = I.

    If A^m = I this implies A^m ~ I.

    Claim: if X~Y implies X^n ~ Y^n for n a positive integer.
    By induction.

    Base case n = 2.
    If X~Y then there exists a Z such that X = ZY(Z^-1).
    X^2 = X*X = ZY(Z^-1)ZY(Z^-1) = Z*(Y^2)*(Z^-1).

    Inductive step Assume true for some n.

    X^(n+1) = (X^n) * X = Z(Y^n)(Z^-1)* ZY(Z^-1) = Z (Y^(n+1))(Z^-1) and we are done.

    So since A^m ~ I implies A^(m+1) ~ I ^2 implies A^(m+1) ~ I, but since A^(m+1) = A we have A~I and since I is a diagonal matrix we are done.
     
  2. jcsd
  3. Dec 11, 2008 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    No, that doesn't work. You've shown X~Y implies X^n~Y^n. That's fine. But applying that to A^m~I only gives you e.g. (A^m)^2~I^2 or A^(2m)~I. That doesn't help. Think about what the blocks must look like in the Jordan normal form of A.
     
    Last edited: Dec 11, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Similar matrices please check my proof.
Loading...