Help finding Eigenvectors

Question

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?

Answer 1

You're correct. The eigenvalues are $\lambda_1=1,\lambda_2=1-2i,\lambda_3=1+2i$

$$v_1=\left[\begin{array}{rrr|r} 0 & 0 & 0 & 0 \\ 2 & 0 & -2 & 0 \\ 3 & 2 & 0 & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & -1 & 0 \\ 0 & 1 & \frac{3}{2} & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} x_3 \\ -\frac{3}{2}\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 1 \\ -\frac{3}{2} \\ 1 \\ \end{array}\right]$$

$$v_2=\left[\begin{array}{rrr|r} 2i & 0 & 0 & 0 \\ 2 & 2i & -2 & 0 \\ 3 & 2 & 2i & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & i & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} 0 \\ -i\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 0 \\ -i \\ 1 \\ \end{array}\right]$$

$$v_3=\left[\begin{array}{rrr|r} -2i & 0 & 0 & 0 \\ 2 & -2i & -2 & 0 \\ 3 & 2 & -2i & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & -i & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} 0 \\ i\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 0 \\ i \\ 1 \\ \end{array}\right]$$