From the spectrum theorem, we know real symmetric matrices have real eigenvalues.

But can real non-symmetric matrix have real eigenvalues?

What are the necessary and sufficient conditions for a real matrix to have real eigenvalues?


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  • $\begingroup$ @usεr11852 but what is the necessary and sufficient condition? $\endgroup$ – hxd1011 Jan 17 '18 at 22:15
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    $\begingroup$ Here's a short paper claiming sufficiency conditions: ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1448531 $\endgroup$ – Moss Murderer Jan 17 '18 at 22:21
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    $\begingroup$ Just conjugate any real diagonal matrix by any invertible but non-orthogonal matrix to produce an example. Here's minimal R code: a <- matrix(c(1,1,0,1),2); b <- a %*% diag(c(2,-1)) %*% solve(a); eigen(b) You will find that the non-symmetric matrix b has real eigenvalues $2,-1$ as specified. $\endgroup$ – whuber Jan 17 '18 at 22:59

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