Eigenvalue of non-symmetric real matrix

Question

Let $A\in M_n(\mathbb{R})$ be a non-symmetric matrix.

Prove that $\lambda_1$ is real, provided that $|\lambda_1|>|\lambda_2|≥|\lambda_3|≥...≥|\lambda_n|$ where $\lambda_i$, $i= 1,...,n$ are the eigenvalues of $A$, while others can be real or not real.

Please give me some hints

Answer 1

Hint: Presumably you mean "Prove that $\ \lambda_1\ $ is real", since $\ \left\vert z\right\vert\ $ is always real for any complex or real $\ z\ $.

The fact that the inequality $\ \lambda_1 > \left\vert \lambda_2\right\vert\ $ is a strict inequality is significant. Keeping Fabian's question in mind, would that inequality be possible if $\ \lambda_1\ $ were complex?

Answer 2

Hint: A real matrix has a real characteristic polynomial. Non-real roots of a real polynomial appear always in complex conjugated pairs. Complex conjugated numbers have equal absolute values. What does it say about a root, which absolute value has no pair?