How to prove O is compact in M?

**whitacre** · June 2nd 2011, 01:10 PM

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?

**girdav** · June 2nd 2011, 01:29 PM

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.

**Tinyboss** · June 3rd 2011, 06:50 AM

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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