Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Show that the set of all orthogonal matrices in the set of all n cross n matrices endowed with any norm topology is complact

share|improve this question
The column vectors of an orthogonal matrix are unit vectors. And there are $n$ column vectors. –  PEV Jan 23 '11 at 21:28
It would also be expeditious to use the operator norm. What is the operator norm of an orthogonal matrix? –  hardmath Jan 23 '11 at 21:42

1 Answer 1

Recall a compact subset of $R^{n \times n}$ is a set that is closed and bounded. One way to show closedness is to observe that the orthogonal matrices are the inverse image of the element $I$ under the continuous map $M \rightarrow MM^T$. Boundedness follows for example from the fact that each column or row is a vector of magnitude $1$.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.