Basis consisting of eigenvectors

Question

How to find a basis consisting of eigenvectors for the following matrix? $$A=\begin{bmatrix} 3 & -1 & -1 \\-1 & 3 & -1 \\ -1 & -1 & 3 \end{bmatrix}$$ Any tips would be great, thanks.

Answer 1

HINT

• compute eigenvalues by $\det(A-\lambda I)=0$
• for each $\lambda$ solve $(A-\lambda I)x=0$ to determine eigenvectors $x$

Note that since the matrix is real symmetric we can find an orthogonal basis of eigenvectors.

Answer 2

A first guess is that all three rows add up to $1$, so $\left[\begin{array}{r}1&1&1\end{array}\right]^{\text{T}}$ is an eigenvector. Thus, the other two eigenvertors must be orthogonal to it (i.e., their entries must add up to zero). Such guesses as $\left[\begin{array}{r}1&-1&0\end{array}\right]^{\text{T}}$ and $\left[\begin{array}{r}1&1&-2\end{array}\right]^{\text{T}}$ would do.