Question involving an invertible complex matrix.

Question

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 $\mathbb{R}$.

Answer 1

Consider the complex polynomial $\lambda \mapsto \det(R + \lambda J)$. Note that it is non-zero when $\lambda = i$, so it is not the zero polynomial. Thus it has finitely many $0$s in the complex plane, which means we cannot have $\det(R + \lambda J) = 0$ for all $\lambda \in \Bbb{R}$.

Now, suppose that $A$ and $B$ are real matrices, and $P = R + iJ$ is a complex invertible matrix such that $$B = P^{-1}AP \iff PB = AP \iff RB + iJB = AR + iAJ.$$ Equating real and imaginary parts, \begin{align*} RB &= AR \\ JB &= AJ. \end{align*} By the previous exercise, we can find some $\lambda_0 \in \Bbb{R}$ such that $Q = R + \lambda_0 J$ is invertible. We therefore have, $$QB = RB + \lambda_0 JB = AR + \lambda_0 AJ = AQ \implies B = Q^{-1} AQ,$$ proving real-similarity.

Answer 2

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.