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.
Add to solve later 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.
Add to solve later 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.
Add to solve later 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 $A$.
Add to solve later 
