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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

    HallsofIvy

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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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