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The center of GL(n)

  1. Apr 4, 2008 #1
    Is the center of GL(n) the set diagonal martices? It's easy to prove that ADA^-1 = D if D is diagonal, but if ABA^-1 = B is true that B is diagonal.
     
  2. jcsd
  3. Apr 4, 2008 #2
    no, the center is the set of all scalar matrices, ie, A is in the center iff A = c*I for some c in F\{0}

    consider elementary matrices


    another suggestion, when working on problems like this where you don't know what it is you want to prove, think simple, ie, take n = 2, and A = (1 0; 0 -1) which is diagonal, and B = (2 3; 1 1), then AB != BA, so A is not in the center but A is a diagonal matrix.
     
    Last edited: Apr 4, 2008
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