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Homework Help: Continuous functions of nxn invertible matrices

  1. Mar 23, 2009 #1
    Ok, so this was assigned as a bonus problem in my Topology class a while ago. Nobody in the class got it, but I've still been racking my brain on it ever since.
    For some n, consider the set of all nxn nonsingular matrices, and using the usual Euclidean topology on this space, show that:

    a) the inverse is a continuous function. f(A) -> A^(-1)

    b) matrix multiplication is a continuous function. g(A,B) -> AB


    I've thought quite a bit about this problem, but I really don't know where to go. If I can show that the determinant is continuous, then I think I can do part a, since calculating the inverse is just equivalent to calculating a bunch of determinants (Cramer's rule).

    As for part b, I am lost.
  2. jcsd
  3. Mar 23, 2009 #2


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    What IS the "usual Euclidean topology" on the set of n by n nonsingular matrices?
  4. Mar 23, 2009 #3
    Treat the matrices as vectors in [tex]\mathbb{R}^{n^{2}}[/tex]

    So, [tex]d(A,B)=\left\| A-B \right\| = \sqrt{\sum_{i, j}\left( a_{i, j}-b_{i, j} \right)^{2}}[/tex]
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