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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- Jun 2nd 2011, 01:10 PMwhitacreHow 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? - Jun 2nd 2011, 01:29 PMgirdav
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.

- Jun 3rd 2011, 06:50 AMTinyboss
The set of $\displaystyle k\times n$ matrices can by identified in the obvious way with $\displaystyle \mathbb{R}^{nk}$, 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 $\displaystyle A\mapsto AA^T$ continuous?