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

Powers of matrices equal to the identity matrix

  1. Nov 18, 2012 #1
    I am curious about under what conditions the powers of a square matrix can equal the identity matrix.

    Suppose that A is a square matrix so that [itex] A^{2} = I [/itex]

    At first I conjectured that A is also an identity matrix, but I found a counterexample to this.
    I noticed that the counterexample was an elementary matrix.

    So then I conjectured that A is an elementary matrix. Is this true? Can I prove this? What about for general powers of A?

    BiP
     
  2. jcsd
  3. Nov 18, 2012 #2

    AlephZero

    User Avatar
    Science Advisor
    Homework Helper

    As a simple example think about 2x2 matrices.

    If ##\displaystyle A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}##, then

    ##\displaystyle A^2 = \begin{bmatrix}a^2 + bc & b(a+d) \\ c(a+d) & d^2 + bc \end{bmatrix} = I##.

    From the off-diagonal terms, ##b(a+d) = 0## and ##c(a+d) = 0##.
    Taking ##b = c = 0## isn't going to lead to anywhere interesting, so let's see what happens if ##d = -a##.
    From the diagonal terms, ##a^2 + bc = 1##.

    You can satisfy that with matrices that are not elementary, for example
    ##\displaystyle A = \begin{bmatrix} 2 & 3 \\ -1 & -2 \end{bmatrix}##.

    In fact the condition ##a^2 + bc = 1## here is the same as ##|\det A| = 1##, which isn't a complete coincidence - but things are not so simple for bigger matrices.
     
  4. Nov 18, 2012 #3
    I see. Thanks much [itex]\aleph_0[/itex]

    BiP
     
  5. Nov 19, 2012 #4
    A solution to An=I is obviously attained if A is a suitable diagonal or rotation matrix, and also for all similar matrices PAP-1, where P is invertible.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Powers of matrices equal to the identity matrix
Loading...