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Upper Triangular Matrix

  1. Feb 26, 2006 #1
    "Let A be an upper triangular matrix with entires in a field F. Suppose that all the diagonal entries of A are equal. Show that A is diagonalizable if and only if it is diagonal."

    I'm reviewing old assignments for a midterm. I remember doing this (backward direction is trivial), I can't remember how, but I remember it was easy. Any hints?
     
  2. jcsd
  3. Feb 26, 2006 #2

    shmoe

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    If A is similar to a diagonal matrix D, what can you say about the diagonal entries of D?
     
  4. Feb 26, 2006 #3
    What do you mean by similar? Do you mean they differ by a few entries? In that case the diagonal entries of D will be the same of that of A.
     
  5. Feb 27, 2006 #4

    shmoe

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    A is similar to D if there exists an invertible matrix X where X*A*X^{-1}=D.

    Saying "A is similar to a diagonal matrix D" is the same thing as saying "A is diagonalizable" except I find it gives a less cumbersome way to give this diagonal matrix a name.

    So what can you say about the entries of this D?
     
  6. Feb 27, 2006 #5
    I don't know, but I found a way of doing it:

    A is upper triangular, and the diagonal entries are all equal, therefore A has a single eingenvalue e. Suppose A is diagonalizable, then V=null(A-eI), therefore A=eI.
     
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