# Help finding Eigenvectors

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?

• Your $v_2$ and the book's $v_2$ differ by a scalar multiple, so they're both eigenvectors for $\lambda_2$. Similarly for $v_3$. So you and the book are both correct. Feb 21, 2021 at 2:51
• Looks absolutely fine to me. As the previous comment noted, eigenvectors for one-dimensional eigenspaces are only defined up to a scalar multiple. If the eigenspace is larger than that, you're just pulling two or more vectors out of a large space and I would generally not expect software to necessarily give the same answer. Feb 21, 2021 at 3:04
• Ah I see now, thank you. Feb 21, 2021 at 3:05

You're correct. The eigenvalues are $$\lambda_1=1,\lambda_2=1-2i,\lambda_3=1+2i$$
$$v_1=\left[\begin{array}{rrr|r} 0 & 0 & 0 & 0 \\ 2 & 0 & -2 & 0 \\ 3 & 2 & 0 & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & -1 & 0 \\ 0 & 1 & \frac{3}{2} & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} x_3 \\ -\frac{3}{2}\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 1 \\ -\frac{3}{2} \\ 1 \\ \end{array}\right]$$
$$v_2=\left[\begin{array}{rrr|r} 2i & 0 & 0 & 0 \\ 2 & 2i & -2 & 0 \\ 3 & 2 & 2i & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & i & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} 0 \\ -i\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 0 \\ -i \\ 1 \\ \end{array}\right]$$
$$v_3=\left[\begin{array}{rrr|r} -2i & 0 & 0 & 0 \\ 2 & -2i & -2 & 0 \\ 3 & 2 & -2i & 0 \\ \end{array}\right]\sim\left[\begin{array}{rrr|r} 1 & 0 & 0 & 0 \\ 0 & 1 & -i & 0 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]\longrightarrow\left[\begin{array}{rrr|r} 0 \\ i\cdot x_3 \\ x_3 \\ \end{array}\right]=\left[\begin{array}{rrr|r} 0 \\ i \\ 1 \\ \end{array}\right]$$