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Let's assume that real $A,B$ are similar over $\mathbb{C}$. Is it true that $A, B$ are similar over $\mathbb{R}$?

The first step should look like this: $P=U+iV$, we have $PA=BP$, then $(U+iV)(A)=B(U+iV)$, $UA+iVA=BU+iBV$, according to the fact that $A, B, U, V$ are real, we state that $UA=BU, VA=BV$, $A=U^{-1}BU, A=V^{-1}BV$.

Actually, the solution above doesn't work if $U, V$ are not invertible. According to the fact that $P$ is invertible and $P=U+iV$ --- how to prove that $U, V$ are also invertible?

hyperkahler
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1 Answers1

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You don't need that $U$ and $V$ are invertible, all you need is that $U + iV$ is invertible because that implies that $U + cV$ is invertible for some real valued $c$. Then that value of $c$ gives you the real valued matrix you want.

user2566092
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