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Matrices satisfying certain relations

  1. Nov 21, 2012 #1
    How do you find matrices a,b,c satisfying
    a=b*c*b^-1
    b=c*a*c^-1
    c=a*b*a^-1 ?
     
  2. jcsd
  3. Nov 21, 2012 #2
    If you know what's diagonalization, you can skip this.


    For a to be diagonalizable, A=PDP^-1, where

    P is an invertible matrix whose columns are A's eigenvector (order of these columns doesn't matter). C is a diagonal matrix that has all A's eigenvalues

    So for a 3x3 diagonalizable matrix
    D=
    λ1 0 0
    0 λ2 0
    0 0 λ3

    λ{1,2,3} are A's eigenvalues

    P=
    [v1 v2 v3]
    v{1,2,3} are A's eigenvectors

    From those 3 equations in your post you can see that a, b and c have to be all diagonal matrices.

    Also, a has to have b's eigenvalues, b has to have c's eigenvalues and c has to have a's eigenvalues. And of course, a has to have c's eigenvectors... etc

    Not sure how i would start solving this, but I hope this helps.
     
  4. Nov 22, 2012 #3
    Hi Aija, your statement above is just wrong. From those 3 equations, you should immeditately observe the solution a=b=c=M, where M is any invertible matrix, and the "problem" is to determine the remaining solutions, if any.
     
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