Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.
Compactness of the set of $n \times n$ orthogonal matrices
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$\begingroup$
matrices
compactness
orthogonal-matrices
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2The column vectors of an orthogonal matrix are unit vectors. And there are $n$ column vectors. – 2011-01-23
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0It would also be expeditious to use the operator norm. What is the operator norm of an orthogonal matrix? – 2011-01-23
1 Answers
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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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0Can you elaborate on how this implies boundedness? – 2014-12-12
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0Each entry must be of absolute value at most 1, since the column it is in has magnitude 1, for example. – 2014-12-12
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0Yes but what is the norm? – 2014-12-12
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1"Magnitude 1" means Euclidean norm here. – 2014-12-12
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1Why is the map continuous? Can you explain it a little bit more? I am thinking of a clear and concise way to prove it. (I have background in functional analysis.) – 2017-04-22