I have the matrix
$$A = \begin{bmatrix}
4 & 2 & 2\\
2 & 4 & 2\\
2 & 2 & 4
\end{bmatrix}$$
I've calculated the Eigenvalues and Eigenvectors as follows with help in a previous question:
$\lambda = 2$ or $8$
$$E_{\lambda = 2} = \left\{ \begin{bmatrix} v_1\\ v_2\\ v_3\\ \end{bmatrix} = a\begin{bmatrix} -1\\ 1\\ 0\\ \end{bmatrix} + b \begin{bmatrix} -1\\ 0\\ 1 \end{bmatrix} \mid a, b \in \mathbb R \right\}$$
$$E_{\lambda = 8} = \left\{ \begin{bmatrix} v_1\\ v_2\\ v_3\\ \end{bmatrix} = a \begin{bmatrix} 1\\ 1\\ 1\\ \end{bmatrix} \mid \mathbb R\right\}$$
Is the Eigenbasis simply:
$$E_{\lambda = 2} = \operatorname{span}\left(\begin{bmatrix} -1\\ 1\\ 0\\ \end{bmatrix} \begin{bmatrix} -1\\ 0\\ 1\\ \end{bmatrix} \right) $$
$$E_{\lambda = 8} = \operatorname{span}\left(\begin{bmatrix} 1\\ 1\\ 1\\ \end{bmatrix}\right)$$
Best Answer
An eigenbasis is a basis in which every vector is an eigenvector.
In your case, $$ \left\{ \pmatrix{-1\\1\\0}, \pmatrix{-1\\0\\1}, \pmatrix{1\\1\\1} \right\} $$ is an eigenbasis for your matrix $A$.