Consider the matrix \begin{equation*} F= \begin{bmatrix} -59 & -34 & 41 & 12 & 25 & 30\\ 1 & 7 & -46 & -36 & -11 & -29\\ -233 & -119 & 58 & -35 & 75 & 54\\ 157 & 81 & -43 & 21 & -51 & -39\\ -91 & -48 & 32 & -5 & 32 & 26\\ 209 & 107 & -55 & 28 & -69 & -50 \end{bmatrix} \end{equation*} then \begin{align*} \charpoly{F}{x}&=-50+55x+13x^2-50x^3+32x^4-9x^5+x^6\\ &=(x-2)(x+1)(x^2-4x+5)^2\\ &=(x-2)(x+1)((x-(2+i))(x-(2-i)))^2\\ &=(x-2)(x+1)(x-(2+i))^2(x-(2-i))^2\\ \end{align*} So the eigenvalues are $\lambda=2,\,-1,2+i,\,2-i$ with algebraic multiplicities $\algmult{F}{2}=1$, $\algmult{F}{-1}=1$, $\algmult{F}{2+i}=2$ and $\algmult{F}{2-i}=2$.
We compute eigenvectors, noting that the last two basis vectors are each a scalar multiple of what Theorem BNS will provide, \begin{align*} \lambda&=2\quad\quad F-2I_6=\\ & \begin{bmatrix} -61 & -34 & 41 & 12 & 25 & 30\\ 1 & 5 & -46 & -36 & -11 & -29\\ -233 & -119 & 56 & -35 & 75 & 54\\ 157 & 81 & -43 & 19 & -51 & -39\\ -91 & -48 & 32 & -5 & 30 & 26\\ 209 & 107 & -55 & 28 & -69 & -52 \end{bmatrix} \rref \begin{bmatrix} \leading{1} & 0 & 0 & 0 & 0 & \frac{1}{5}\\ 0 & \leading{1} & 0 & 0 & 0 & 0\\ 0 & 0 & \leading{1} & 0 & 0 & \frac{3}{5}\\ 0 & 0 & 0 & \leading{1} & 0 & -\frac{1}{5}\\ 0 & 0 & 0 & 0 & \leading{1} & \frac{4}{5}\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}\\ &\eigenspace{F}{2}=\nsp{F-2I_6} =\spn{\set{\colvector{-\frac{1}{5}\\0\\-\frac{3}{5}\\\frac{1}{5}\\-\frac{4}{5}\\1}}} =\spn{\set{\colvector{-1\\0\\-3\\1\\-4\\5}}}\\ \end{align*} \begin{align*} \lambda&=-1\quad\quad F+1I_6=\\ & \begin{bmatrix} -58 & -34 & 41 & 12 & 25 & 30\\ 1 & 8 & -46 & -36 & -11 & -29\\ -233 & -119 & 59 & -35 & 75 & 54\\ 157 & 81 & -43 & 22 & -51 & -39\\ -91 & -48 & 32 & -5 & 33 & 26\\ 209 & 107 & -55 & 28 & -69 & -49 \end{bmatrix} \rref \begin{bmatrix} \leading{1} & 0 & 0 & 0 & 0 & \frac{1}{2}\\ 0 & \leading{1} & 0 & 0 & 0 & -\frac{3}{2}\\ 0 & 0 & \leading{1} & 0 & 0 & \frac{1}{2}\\ 0 & 0 & 0 & \leading{1} & 0 & 0\\ 0 & 0 & 0 & 0 & \leading{1} & -\frac{1}{2}\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}\\ &\eigenspace{F}{-1}=\nsp{F+I_6} =\spn{\set{\colvector{-\frac{1}{2}\\\frac{3}{2}\\-\frac{1}{2}\\0\\\frac{1}{2}\\1}}} =\spn{\set{\colvector{-1\\3\\-1\\0\\1\\2}}}\\ \end{align*} \begin{align*} \lambda&=2+i\\ &F-(2+i)I_6= \begin{bmatrix} -61-i & -34 & 41 & 12 & 25 & 30\\ 1 & 5-i & -46 & -36 & -11 & -29\\ -233 & -119 & 56-i & -35 & 75 & 54\\ 157 & 81 & -43 & 19-i & -51 & -39\\ -91 & -48 & 32 & -5 & 30-i & 26\\ 209 & 107 & -55 & 28 & -69 & -52-i \end{bmatrix}\\ & \rref \begin{bmatrix} \leading{1} & 0 & 0 & 0 & 0 & \frac{1}{5}(7+ i)\\ 0 & \leading{1} & 0 & 0 & 0 & \frac{1}{5}(-9-2i)\\ 0 & 0 & \leading{1} & 0 & 0 & 1\\ 0 & 0 & 0 & \leading{1} & 0 & -1\\ 0 & 0 & 0 & 0 & \leading{1} & 1\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}\\ &\eigenspace{F}{2+i}=\nsp{F-(2+i)I_6} =\spn{\set{\colvector{-7-i\\9+2i\\-5\\5\\-5\\5}}}\\ \end{align*} \begin{align*} \lambda&=2-i\\ &F-(2-i)I_6= \begin{bmatrix} -61+i & -34 & 41 & 12 & 25 & 30\\ 1 & 5+i & -46 & -36 & -11 & -29\\ -233 & -119 & 56+i & -35 & 75 & 54\\ 157 & 81 & -43 & 19+i & -51 & -39\\ -91 & -48 & 32 & -5 & 30+i & 26\\ 209 & 107 & -55 & 28 & -69 & -52+i \end{bmatrix}\\ &\rref \begin{bmatrix} \leading{1} & 0 & 0 & 0 & 0 & \frac{1}{5}(7-i)\\ 0 & \leading{1} & 0 & 0 & 0 & \frac{1}{5}(-9+2i)\\ 0 & 0 & \leading{1} & 0 & 0 & 1\\ 0 & 0 & 0 & \leading{1} & 0 & -1\\ 0 & 0 & 0 & 0 & \leading{1} & 1\\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}\\ &\eigenspace{F}{2-i}=\nsp{F-(2-i)I_6} =\spn{\set{\colvector{-7+i\\9-2i\\-5\\5\\-5\\5}}}\\ \end{align*}
Eigenspace dimensions yield geometric multiplicities of $\geomult{F}{2}=1$, $\geomult{F}{-1}=1$, $\geomult{F}{2+i}=1$ and $\geomult{F}{2-i}=1$. This example demonstrates some of the possibilities for the appearance of complex eigenvalues, even when all the entries of the matrix are real. Notice how all the numbers in the analysis of $\lambda=2-i$ are conjugates of the corresponding number in the analysis of $\lambda=2+i$. This is the content of the upcoming Theorem ERMCP.