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We can prove that diagonalizable matrices with complex values are dense in set of $n \times n$ complex matrices, as it was previously answered.

I don't think there is many chances but can we show that the set of diagonalizable matrices over $\mathbb R$ with real entries is dense in $M_n(\mathbb R)$?

anni
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    This help? https://math.stackexchange.com/questions/2184461/the-diagonalizable-matrices-are-not-dense-in-the-square-real-matrices – T. Fo Dec 13 '18 at 22:30
  • I believe a rotation in $\mathbb{R}^2$ namely $((0,1),(-1,0))$ can not be approximated by diagonal matrices, since it has eigenvalue $i$. – SmileyCraft Dec 13 '18 at 22:30

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