I need to solve the following problem:
The set of all ortogonal matrix $ n \times n $ is disconnected.
I would really appreciate your help with this problem thank you.
2026-04-04 00:16:24.1775261784
the set of all orthogonal matrix is disconnected
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let $ det : \mathbb M_{n \times n} \longrightarrow \mathbb R $ be the determinant function. This function is continuons and then carries connected sets in connected sets. The restriction of this function in the set of all ortogonal matrix is such that this image is the set $\{ -1, 1 \}$. But this set is disconnected.
I appreciate the suggestion given by the user Ian. Your suggestion helped me to solve this problem. thank you so much!