# Compactness of orthogonal Matrices

Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.

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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

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 $1$.

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Can you elaborate on how this implies boundedness? – learner Dec 12 '14 at 2:38
Each entry must be of absolute value at most 1, since the column it is in has magnitude 1, for example. – Zarrax Dec 12 '14 at 5:02
Yes but what is the norm? – learner Dec 12 '14 at 5:21
"Magnitude 1" means Euclidean norm here. – Zarrax Dec 12 '14 at 14:07