Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Raising a bunch of matrices to a power

  1. Dec 10, 2009 #1
    Let A, B, and C be nxn matrices,
    I'm wondering
    1. is it possible to simplify (ABC)^5 to expand it ( maybe into something like (A^5)(B^5)(C^5) )
    2. what's the fastest way of solving (ABC)^5? I'm thinking actually multiply ABC out, then diagonalize it. Is there a faster way?
  2. jcsd
  3. Dec 10, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper

    Under certain conditions. If A, B and C commute among themselves (that is, AB = BA, AC = CA and BC = CB) then this is possible.

    I'm thinking the same thing. Alternatively you could try writing down an expression in terms of A^5, B^5, C^5 and the commutators [A, B] = AB - BA, [A, C] = AC - CA and [B, C] = BC - CB, e.g.
    (A B C)^2 = A B C A B C
    ... = A B A C B C + A B [C, A] B C
    ... = A^2 B C B C + A [B, A] C B C + A B [C, A] B C
    ... = A^2 B^2 C^2 - A^2 B [B, C] C - A [A, B] C B C - A B [A, C] B C

    But that is in general ugly and not very helpful.
  4. Dec 10, 2009 #3
    Thank you CompuChip
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook