The matrix is \begin{equation*}
A =
\begin{pmatrix}
1 & 0 & 0 \\
2 & 1 & -2 \\
3 & 2 & 1
\end{pmatrix}
\end{equation*}
I got the eigenvalues $\lambda_1 = 1, \lambda_2 = 1 + 2i$, and $\lambda_3 = 1-2i$. I am only concerned with the complex valued eigenvectors. For $\lambda_2$, I got the eigenvector
\begin{equation*} v_2=
\begin{pmatrix}
0 \\
i \\
1
\end{pmatrix}\end{equation*} and for $\lambda_3$, I got the eigenvector
\begin{equation*}v_3=
\begin{pmatrix}
0 \\
-i \\
1
\end{pmatrix}\end{equation*}
In the back of the book, it is saying the eigenvectors for $\lambda_2$ and $\lambda_3$ are \begin{equation*} v_2=
\begin{pmatrix}
0 \\
1 \\
-i
\end{pmatrix}\end{equation*}
and \begin{equation*}v_3=
\begin{pmatrix}
0 \\
1 \\
i
\end{pmatrix}\end{equation*}
When I checked on Wolfram Alpha, it is saying that my answers are correct. Did I do something wrong or is the back of my book wrong?
Best Answer
You're correct. The eigenvalues are $\lambda_1=1,\lambda_2=1-2i,\lambda_3=1+2i$
$$v_1=\left[\begin{array}{rrr|r} 0 & 0 & 0 & 0 \\ 2 & 0 & -2 & 0 \\ 3 & 2 & 0 & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & -1 & 0 \\ 0 & 1 & \frac{3}{2} & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} x_3 \\ -\frac{3}{2}\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 1 \\ -\frac{3}{2} \\ 1 \\ \end{array}\right]$$
$$v_2=\left[\begin{array}{rrr|r} 2i & 0 & 0 & 0 \\ 2 & 2i & -2 & 0 \\ 3 & 2 & 2i & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & i & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} 0 \\ -i\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 0 \\ -i \\ 1 \\ \end{array}\right]$$
$$v_3=\left[\begin{array}{rrr|r} -2i & 0 & 0 & 0 \\ 2 & -2i & -2 & 0 \\ 3 & 2 & -2i & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & -i & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} 0 \\ i\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 0 \\ i \\ 1 \\ \end{array}\right]$$