# Finding Eigenvectors of a $3\times3$ Matrix (7.12-15)

Please check my work in finding an eigenbasis (eigenvectors) for the following problem. Some of my solutions do not match answers in my differential equations text (Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons).

For reference the following identity is given because some textbooks reverse the formula having $$\lambda$$ subtract the diagonal elements instead of subtracting $$\lambda$$ from the diagonal elements:

$$\det(A - \lambda I) = 0$$ $$A = \begin{bmatrix} 3 & 1 & 4 \\ 0 & 2 & 6 \\ 0 & 0 & 5 \end{bmatrix}$$

By inspection the eigenvalues are the entries along the diagonal for this upper triangular matrix. $$\lambda_1 = 3 \qquad \lambda_2 = 2 \qquad \lambda_3 = 5$$ When $$\lambda_1 = 3$$ we have: $$A - 3I = \begin{bmatrix} 3-3 & 1 & 4 \\ 0 & 2-3 & 6 \\ 0 & 0 & 5-3 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 4 \\ 0 & -1 & 6 \\ 0 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ $$x_1 = 1 \ \text{(free variable)} \qquad x_2 = 0 \qquad x_3 = 0 \\$$ $$v_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \qquad \text{(matches answer in text)}$$ When $$\lambda_2 = 2$$ we have: $$A - 2I = \begin{bmatrix} 3-2 & 1 & 4 \\ 0 & 2-2 & 6 \\ 0 & 0 & 5-2 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 4 \\ 0 & 0 & 6 \\ 0 & 0 & 3 \end{bmatrix} = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ $$x_1 = -x_2 \qquad x_2 = 1 \ \text{(free variable)} \qquad x_3 = 0 \\$$ $$v_2 = \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} \qquad \text{but answer in text is} \qquad \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$$ What happened? Is it from a disagreement in what we should consider arbitrary or am I doing something fundamentally wrong?

When $$\lambda_3 = 5$$ we have: $$A - 5I = \begin{bmatrix} 3-5 & 1 & 4 \\ 0 & 2-5 & 6 \\ 0 & 0 & 5-5 \end{bmatrix} = \begin{bmatrix} 2 & 1 & 4 \\ 0 & -3 & 6 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & -3 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{bmatrix}$$ $$x_1 = 3x_3 \qquad x_2 = 2x_3 \qquad x_3 = 1 \ \text{(free variable)} \\$$ $$v_3 = \begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix} \qquad \text{(matches answer in text)}$$

• Both your and the text book's answer work. If $(\lambda ,v)$ is an eigenpair of a matrix $M$, then so is $(\lambda, \mu v)$, for all scalars $\mu$. Jan 23, 2016 at 23:15

Eigenvectors are never unique. In particular, for the eigenvalue $2$ you can take, for example, $x_2=-1$ which gives you the answer in the book.
• Precisely, and in fact you could take for $x_2$ any nonzero real number. Jan 23, 2016 at 23:26