Help with eigenvectors of an operator

Question

We have the following operator

$\hat{A} = 2|u_{1}\rangle \langle u_{1}| + 2|u_{2}\rangle \langle u_{2}| + 1|u_{3}\rangle \langle u_{3}|$

with $|u_{i}\rangle $ an orthonormal base. The matrix representation of this operator is

\begin{pmatrix} 2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{pmatrix}

the eigenvalues are {2, 2, 1}, and the corresponding orthonormal eigenvectors are {(1 0 0), (0 1 0), (0 0 1)}

Is it possible to find a non orthonormal base of eigenvectors?

Answer 1

Since any vector of the form $(a,b,0)$ is an eigenvector with eigenvalue $2$, you can take, for instance, the basis $\{(1,0,0),(1,1,0),(0,0,1)\}$.