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Math Help - Orthognal matrices are compact

  1. #1
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    Orthognal matrices are compact

    HI,
    How can I prove the set of compact matrices are compact
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  2. #2
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    Do you mean Orthogonal matrices are compact You wrote:
    How can I prove the set of compact matrices are compact
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  3. #3
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    Orthognal matrices are compact

    Yes.Set of orthogonal matrices are compact.I now why they are bounded but does not know why they are closed.
    Last edited by Math Help; June 5th 2005 at 09:11 AM.
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    Please give us an equation to define the matrix.
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  5. #5
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    Quote Originally Posted by Math Help
    Do you mean Orthogonal matrices are compact You wrote:
    How can I prove the set of compact matrices are compact
    Let O = set of orthogonal nxn matrices, that is O = {A | A'A = E}, where A' = transpose of A and E = unit matrix.

    The condition A'A = E actually is a set of n^2 equations that have to hold. The left hand side of each equation is a continuous function on the set of all nxn matrices (just a quadratic form, actually). The right hand side of each equation is either 0 or 1, a closed set.

    Preimages of closed sets under continuous functions are closed.

    Finite intersections of closed sets are closed.

    Done: O is closed. (You already proved it's bounded).
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    Can you explain how you know they are bounded? Thanks.
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  7. #7
    hpe
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    Quote Originally Posted by deannalouise View Post
    Can you explain how you know they are bounded? Thanks.
    Trace(O'O) = n. That's the sum of squares of the entries of the matrix O.
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