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Suppose I have a matrix whose entries are in $\mathbb{C}$.

How easy or hard is it to tell in general if a matrix $M$ is similar to a real matrix?

Rasmus pointed out in the comments that in general a matrix has no similar real counterpart. What about for simple matrices like diagonal ones?

Mark
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    It's not true in general: for $1\times 1$-matrices, similar matrices are equal. Hence $(i)$ cannot be similar to a real matrix. – Rasmus Aug 05 '14 at 19:41

1 Answers1

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A complex matrix is similar to a real matrix iff its Jordan canonical form has the property that the Jordan blocks for non-real eigenvalues can be paired up, so that a block for eigenvalue $\lambda$ corresponds to a block of the same size for eigenvalue $\overline{\lambda}$.

Robert Israel
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  • What do you mean paired up? Like they are the same size? – Mark Aug 05 '14 at 20:25
  • I mean, they can be partitioned into disjoint pairs. In each pair you have two blocks of the same size and for complex conjugate eigenvectors. – Robert Israel Aug 06 '14 at 02:01
  • Supposing I had the Jordan form of a complex matrix in question, and the blocks were partitioned as you say. What is the way to turn it into a real matrix? – Mark Aug 06 '14 at 02:33
  • @RobertIsrael Hi, do you happen to have a proof of this claim? I couldn't find it anywhere. – GSofer Nov 12 '20 at 16:12