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Triangular matrices

  1. Dec 9, 2009 #1

    crd

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    It is stated in almost every linear algebra text i could find that the inverse of a triangular matrix is also triangular, but no proofs accompanied such statements.

    I am convinced that it is the truth, but I have not been able to write anything down that I am satisfied with that doesnt rely on the argument that row operations on the matrix (A|I) to obtain (I|A^{-1}).

    Since this would only be the forward pass(if A is lower triangular) and the backwards pass(if A is upper triangular) and these operations ultimately do not introduce non zero terms above/below the diagonal entries(depending on what A was), thus A^{-1} would be a triangular matrix of the same flavor.

    Has anyone come across anything a little more elegant than simply brute forcing it?
     
  2. jcsd
  3. Dec 9, 2009 #2
    Assume you have a invertible upper triangular matrix. Consider

    [tex]AA^{-1} = I [/tex]

    You can use induction, starting from the last row of A times the last column of [itex]A^{-1}[/itex]. gives you the entry. lower left entry 1.

    Then again take the last row of A and n-1 column of [itex]A^{-1}[/itex]. To be able to get a zero in the identity matrix, (n,n-1) entry of [itex]A^{-1}[/itex] must be zero.
    ....
    Carry on to the upper left corner and you are done.
     
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