Let $\mathbf{x}$ be an eigenvector corresponding to the eigenvalue $\lambda$. Then we have
\[A\mathbf{x}=\lambda \mathbf{x}.\]
Taking the conjugate of both sides, we have
\[\overline{A\mathbf{x}}=\overline{\lambda \mathbf{x}}.\]

Since $A$ is a real matrix, it yields that
\[A\bar{\mathbf{x}}=\bar{\lambda}\bar{\mathbf{x}}. \tag{*}\]
Note that $\mathbf{x}$ is a nonzero vector as it is an eigenvector. Then the complex conjugate $\bar{\mathbf{x}}$ is a nonzero vector as well.
Thus the equality (*) implies that the complex conjugate $\bar{\lambda}$ is an eigenvalue of $A$ with corresponding eigenvector $\bar{\mathbf{x}}$.

Proof 2.

Let $p(t)$ be the characteristic polynomial of $A$.
Recall that the roots of the characteristic polynomial $p(t)$ are the eigenvalues of $A$.
Thus, we have
\[p(\lambda)=0.\]

As $A$ is a real matrix, the characteristic polynomial $p(t)$ has real coefficients.
It follows that
\[\overline{p(t)}=p(\,\bar{t}\,).\]
The previous two identities yield that
\begin{align*}
p(\bar{\lambda})=\overline{p(\lambda)}=\bar{0}=0,
\end{align*}
and the complex conjugate $\bar{\lambda}$ is a root of $p(t)$, and hence $\bar{\lambda}$ is an eigenvalue of $A$.

There is at Least One Real Eigenvalue of an Odd Real Matrix
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.
We give two proofs.
Proof 1.
Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$.
It is a degree $n$ […]

Find the Eigenvalues and Eigenvectors of the Matrix $A^4-3A^3+3A^2-2A+8E$.
Let
\[A=\begin{bmatrix}
1 & -1\\
2& 3
\end{bmatrix}.\]
Find the eigenvalues and the eigenvectors of the matrix
\[B=A^4-3A^3+3A^2-2A+8E.\]
(Nagoya University Linear Algebra Exam Problem)
Hint.
Apply the Cayley-Hamilton theorem.
That is if $p_A(t)$ is the […]

Eigenvalues of a Hermitian Matrix are Real Numbers
Show that eigenvalues of a Hermitian matrix $A$ are real numbers.
(The Ohio State University Linear Algebra Exam Problem)
We give two proofs. These two proofs are essentially the same.
The second proof is a bit simpler and concise compared to the first one.
[…]

Find Eigenvalues, Eigenvectors, and Diagonalize the 2 by 2 Matrix
Consider the matrix $A=\begin{bmatrix}
a & -b\\
b& a
\end{bmatrix}$, where $a$ and $b$ are real numbers and $b\neq 0$.
(a) Find all eigenvalues of $A$.
(b) For each eigenvalue of $A$, determine the eigenspace $E_{\lambda}$.
(c) Diagonalize the matrix $A$ by finding a […]

An Example of a Real Matrix that Does Not Have Real Eigenvalues
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.
Proof.
Let $\lambda$ be an arbitrary eigenvalue of […]