# find the eigenvalues and eigenvectors for the matrix

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?

There are actually infinitely many solutions. For any matrix $$A$$ suppose it has eigenvalue $$\lambda_1$$ with eigenvector $$v_1$$. Then for any scalar $$c\neq 0$$, $$A(cv_1)=c(Av_1)=c(\lambda_1v_1)=\lambda_1(cv_1)$$ so $$cv_1$$ is also a solution. In your case the two solutions differ by a factor of $$-1$$.
To answer your question, we do allow for distinct solutions. If this is for a homework you would only need to write down one of the solutions because the dimension for the eigenspace of $$\lambda_1$$ is $$1$$. You only need one nonzero solution to be able to generate any other solution for this eigenvalue.
As @gt6989b mentioned some matrices have eigenvalues where the dimension of their eigenspace is greater than $$1$$. In this case you would need to provide a set of linearly independent solutions that form a basis for this eigenspace.