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Homework Help: Matrix norm

  1. Jan 9, 2009 #1
    1. The problem statement, all variables and given/known data
    How do I analytically solve for k to satisfy:
    [tex]||kA-I||<1[/tex]?
    Here, k is a real number scalar, A is a known matrix.

    2. Relevant equations

    3. The attempt at a solution
    I am confused because if A was a number, I could break it into two cases where kA-I is positive and negative. But now A is a matrix so I can't do so. How should I look at the problem and what other equations are relevant?

    Suppose the norm of A is 5, e.g. A = [5 0;0 5]. Then I know that the upper limit of k is 2/5, so that kA-I can be at most [1 0;0 1]. I also know that the lower limit of k has to be 0, because if k is ever negative, the norm would be greater than one, since the norm of I is already 1.

    But how do I solve for this kind of result given an arbitrary matrix A?

    - Thanks
     
  2. jcsd
  3. Jan 10, 2009 #2

    HallsofIvy

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    There are several different definitions of "norm" of a matrix. Which are you using here?
     
  4. Jan 10, 2009 #3
    The largest gain in magnitude to a vector:

    For y = Ax, the norm of A is the largest 'a' satisfying |y| = a|x| from all possible choices of x, where |.| is the 2-norm of the vector. Does this definition make sense?
     
  5. Jan 10, 2009 #4

    Dick

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    The vector norm of Ax squared is Ax.Ax (dot product). So you can find the operator norm of A by finding the square root of the largest eigenvalue of A*A^(T).
     
  6. Jan 19, 2009 #5
    So in my original equation where I am trying to find the unknown k such that [tex]||kB-I||<1[/tex], my [tex]A[/tex] is [tex]kB-I[/tex]. To find the eigenvalues of [tex]AA^T[/tex], I do [tex]0 = \lambda I - AA^T = \lambda I - (kB-I)(kB-I)^T[/tex]

    Is there a way to pull out k so that I could solve for k in terms of the old eigenvalues or the old norm?

    - Thanks
     
  7. Jan 19, 2009 #6

    Dick

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    I don't think you can really 'pull out the k' in any useful way. I think you just have to put the matrix B in and crank it out. It could get pretty complicated unless the matrix is small.
     
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