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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?

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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.

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