Is it true that the orthogonal group $O(n)$ spans $M_n(\mathbb{R})$? That is, can every $n$ by $n$ matrix with real coefficients be written as a linear combination of orthogonal matrices?
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$\begingroup$ There is a difference between $SO(n)$ and $O(n)$, which do you mean? $\endgroup$– Mathematician 42May 29, 2017 at 8:50
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$\begingroup$ Just a typo I meant the orthogonal group O(n). $\endgroup$– math_loverMay 29, 2017 at 8:53
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1 Answer
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Hint : To prove it, it suffices to show that $O(n)$ spans the $E_{i,j}$ (matrices that are all $0$'s except in position $(i,j)$, where there's a $1$). For that, consider permutation matrices, and permutation matrices with a few $-$signs
answered May 29, 2017 at 8:55