How do I compute an eigenvector problem when it is underdetermined?

Question

Find the eigenvectors of $$A = \left[ {\begin{array}{ccc} 4 & 0 & 0 \\ -1 & 5 & -1 \\ -1 & 1 & 3 \end{array} } \right]$$

Step 1. Find the eigenvalues.

$$ \left| {\begin{array}{ccc} 4-\lambda & 0 & 0 \\ -1 & 5-\lambda & -1 \\ -1 & 1 & 3-\lambda \end{array} } \right| = (4-\lambda)^3 $$

So the only eigenvalue is $\lambda = 4$, with multiplicity 3.

Step 2. The eigenvector for $\lambda = 4$ is

$$\left[ {\begin{array}{ccc} 0 & 0 & 0 \\ -1 & 1 & -1 \\ -1 & 1 & -1 \end{array} } \right] \left[ {\begin{array}{c} a \\ b \\ c \end{array} } \right] = \left[ {\begin{array}{c} 0 \\ 0 \\ 0 \end{array} } \right] $$

This gives the underdetermined equation system

$$\begin{align*} -a + b -c = 0\end{align*}$$

I don't know how to proceed.

Answer 1

You have two free variables (say $b$ and $c$), so you'll get two linearly independent solutions. For example, you may take $b=1$ and $c=0$ for one and $b=0$ and $c=1$ for the other.

Answer 2

Answer to your question is $[1,2,1]^T$ which is the eigenvector for the eigenvalue 4

$-a+b-c=0 \Rightarrow b=a+c$ Put a=1 and c=1 you get b=2 (a and c are free variables)