# Tagged: real matrix

## Problem 596

Let
$A=\begin{bmatrix} a & b\\ -b& a \end{bmatrix}$ be a $2\times 2$ matrix, where $a, b$ are real numbers.
Suppose that $b\neq 0$.

Prove that the matrix $A$ does not have real eigenvalues.

## Problem 407

Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix.
Prove that the matrix $A$ has at least one real eigenvalue.

## Problem 405

Recall that a complex matrix is called Hermitian if $A^*=A$, where $A^*=\bar{A}^{\trans}$.
Prove that every Hermitian matrix $A$ can be written as the sum
$A=B+iC,$ where $B$ is a real symmetric matrix and $C$ is a real skew-symmetric matrix.

## Problem 404

Let $A$ be an $n\times n$ real matrix.

Prove that if $\lambda$ is an eigenvalue of $A$, then its complex conjugate $\bar{\lambda}$ is also an eigenvalue of $A$.