# If two real matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$? [duplicate]

As in the title:

If two real (square) matrices are conjugated over $\mathbb{C}$, are they then also conjugated over $\mathbb{R}$?

## marked as duplicate by Eric Wofsey, Jyrki LahtonenJul 11 '16 at 21:32

• What does it mean to be conjugate over $\mathbb{R}$? – Morgan Rodgers Jul 11 '16 at 21:22
• @MorganRodgers: I think what's meant by "conjugated over a field $k$" here is "conjugate(d) by conjugation with a matrix with entries in $k$". – joriki Jul 11 '16 at 21:23
• @joriki Ahh, ok so he's most likely asking if the matrices are similar? – Morgan Rodgers Jul 11 '16 at 21:27
• @MorganRodgers: I believe the two terms are essential synonymous; from en.wikipedia.org/wiki/Matrix_similarity, it seems that as far as there's any difference between them, "conjugate" is the more appropriate one. – joriki Jul 11 '16 at 21:32
• See also here for an argument specific to $\Bbb{C}/\Bbb{R}$. – Jyrki Lahtonen Jul 11 '16 at 21:34