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Homework Help: Linear Algebra Problem #3

  1. Jun 30, 2008 #1
    Here we go again.:redface:

    1. The problem statement, all variables and given/known data

    Let S be any nxn strictly upper triangular matrix; prove that [tex]S^n=0[/tex]

    3. The attempt at a solution

    Alright so I know that if c_ij is an entry in a strictly upper triangular matrix, then [tex]c_{ij}= 0, \ i>j[/tex]

    I'll add more in a minute:smile:
     
  2. jcsd
  3. Jun 30, 2008 #2

    Dick

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    Is this another one of those questions where you understand exactly why it's true, but can't phrase it in the form of a proof? Because it is pretty obvious why it's true. Just write down a strictly upper triangular matrix and start taking powers of it. See what happens??
     
  4. Jun 30, 2008 #3

    Defennder

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    I think the problem is with formalising it into matrix entry subscript notation. I can see why it holds: Over time throughout repeated multiplication the non-zero entries of the matrix retreats into the top right hand corner and then finally becomes zero.

    I've written out the matrix entries for an arbitrary c_{ij} after multiplying matrices by 3 times and I'm beginning to see nested summation series which I can't quite simplify.

    PS. I know this is Saladsamurai's problem and not mine, but I'm curious as to how to prove it formally.
     
  5. Jun 30, 2008 #4

    Dick

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    [tex](c^n)_{ij}=c_{i,k_1}*c_{k_1,k_2}*c_{k_2,k_3}*...c_{k_{n-2},k_{n-1}}*c_{k_{n-1},j}[/tex]. All of the k indices summed over. What condition must be true for that to be nonzero? Hey, what ever happened to your electric flux problem? It's not THAT hard.
     
    Last edited: Jun 30, 2008
  6. Jul 1, 2008 #5
    Alternatively, note that the only eigenvalue is 0. Thinking of this matrix as an operator on an n-dimensional vector space V, we must have V = dim null S^(dim V) = dim null S^n, the set of all generalized eigenvectors. So S^n = 0.
     
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