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Math Help - How to prove O is compact in M?

  1. #1
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    How to prove O is compact in M?

    Let M be the k*n matrices of rank k(k <= n),O= { A is in M: A^{t}A=I }.
    How to prove O is compact in M?
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  2. #2
    Super Member girdav's Avatar
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    I guess you mean compact for a topology which is given by a norm. Since M is a finite dimensional vector space, you have to show that O is closed and bounded.
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  3. #3
    Senior Member Tinyboss's Avatar
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    The set of k\times n matrices can by identified in the obvious way with \mathbb{R}^{nk}, and can be given the standard metric topology. Heine-Borel applies in this setting, too. Bounded is easy here, and for closed, here's a hint: is the map A\mapsto AA^T continuous?
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