Question involving an invertible complex matrix. Let $A$ be an $n \times n$ invertible matrix with complex entries and call $A = R + iJ$, where $R$ is the real part of $A$ and $J$ is the imaginary part of $A$. Show that there exist a $\lambda_0 \in \mathbb{R}$ s.t $R+ \lambda_0J$ is invertible. Futhermore, conclude that if $A$ and $B$ are matrices with real entries and they are similar in $\mathbb{C}$ then they are in $\mathbb{R}$.
I've tried to prove the first statement by contradiction, but what kind of things I can do if $\det(R+xJ)= 0$ for each $x \in \mathbb{R}$??
Also, I cannot see the conection with the fact that if $A$ and $B$ are similar in $\mathbb{C}$, then they are in $\mathbb{R}$.

For the first part: as you did, consider $f(x)=\det(R+xJ)$ as a polynomial function (with real coefficients) of some degree $n$ in $x$. You know that $f(i)\ne 0$. So not all coefficients are $0$. But by contradiction, without loss of generality $f(1),f(2),... f(n+1)$ are equal to $0$. There are no non-zero polynomials of degree $n$ with that property, a contradiction. So indeed for some integer $k$ between $1$ and $n+1$, $f(k)\ne 0$ and the matrix $R+kJ$ is invertible.
The second part is not true for every pair or complex $A,B$. You probably missed some conditions.
