Find the eigenvalues and eigenvectors for the matrix A, where $A=\begin{bmatrix} -5 & 6 & 4 \\ -18 & 16 & 8 \\ 72 & -48 & -13 \end{bmatrix}$. There are three distinct eigenvalues for this matrix, but For one of the fundamental solutions, say $\lambda_1$, I found two different kinds eigenvectors $\lambda_1=-5, v_1=(-1,-2,3)^T$
The eigenvector can also be $v_1=(1,2,-3)^T$.
Question: For finding fundamental solutions for eigenvalues and eigenvectors, do we allow those two distinct eigenvectors to coexist?