Is an orthogonal matrix necessarily a permutation matrix?
I believe the answer is no as a permutation matrix is a special case of an orthogonal matrix, but I am having a trouble finding a counterexample. Thanks in advance for your help.
2025-01-13 05:42:53.1736746973
Is an orthogonal matrix necessarily a permutation matrix?
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Hint:
Permutation matrices have only $0$ or $1$ entries.
Orthogonal matrices have columns that are orthogonal unitary vectors.
We can well have orthogonal unitary vectors with entries different from $0$ and $1$. Can you find a simple example in $\mathbb{R}^2$?