Linear Algebra - Give an example for $3x3$ matrix for these eigenvalues

Question

I'm having trouble with this problem :

Give an example for matrix $A$ with these eigenvalues $\lambda_1-1,\lambda_2=1,\lambda_3=0$ while :

$$v_1=(0,1,1)$$ $$v_2=(1,-1,1)$$ $$v_3=(0,1,-1)$$

$v_1,v_2,v_3$ Are eigenvectors of matrix $A$

I don't know how to find matrix $A$ that apply these terms.

The only thing I came up with is that : $$A(\lambda_i)=\lambda_i v_i \rightarrow A(\lambda_i)-\lambda_iv_i =0$$

Any ideas?

Thanks!

Answer 1

Note that $A$ is diagonalizable since $A$ is $3\times 3$ and has three eigenvalues. Hence $$ A=P\Lambda P^{-1} $$ where \begin{align*} P&= \begin{bmatrix} 0 & 1 & 0 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{bmatrix} & \Lambda &= \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix} \end{align*}