Let \(A,B\in M_n(\RR)\). If \(A\) and \(B\) are similar in \(M_n(\CC)\), must they already be similar in \(M_n(\RR)\)?
Solution
Yes.
Suppose there is some \[ X\in GL_n(\CC) \] with \[ AX=XB. \] Write \[ X=P+iQ \] with \(P,Q\in M_n(\RR)\). Since \(A\) and \(B\) are real, comparing real and imaginary parts gives \[ AP=PB, \qquad AQ=QB. \] Hence for every real number \(t\), \[ A(P+tQ)=(P+tQ)B. \]
Consider the real polynomial \[ f(t)=\det(P+tQ). \] It is not identically zero, because \[ f(i)=\det(P+iQ)=\det X\ne0. \] Therefore \(f\) has only finitely many real roots. Choose \[ t\in\RR \] with \(f(t)\ne0\). Then \[ Y=P+tQ\in GL_n(\RR) \] and \[ AY=YB. \] Thus \[ A=YBY^{-1}, \] so \(A\) and \(B\) are already similar over \(\RR\).