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Banach's Lemma

  1. Nov 18, 2013 #1
    1. The problem statement, all variables and given/known data

    This isn't exactly homework per se, i just am having a hard time figuring out how to solve these types of questions generally...

    Consider the 3x3 matrix A = diag[1,3,1]. Show that this matrix is invertible no matter what its dimensions are. Determine an upper bound on [itex]||A^{-1}||_{∞}[/itex] and on the condition number of A

    3. The attempt at a solution

    So from what i've picked up on so far, i start out:

    A = D + E

    where D contains all values of A that are in the same row and column (ie. 1,1... 2,2... etc) and E contains all values of A that are not on the diagonal.

    Next, say that B = [itex]D^{-1}[/itex] * E and A = D(I + B)

    It follows that [itex]||B||_{∞}[/itex] < 1 because of diagonal dominance. Hence A is invertible (I have no idea what diagonal dominance means in this context, i just wrote down what i'm supposed to say in order to get marks here - not much explanation has been given on this part).

    How do i find the condition number and upper bounds?
     
  2. jcsd
  3. Nov 20, 2013 #2
    So apparently the condition number is determined by adding the infinity norm of A with the infinity norm of its inverse - is this correct?
     
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