# Is there any theorem talking about the uniqueness of eigenvector?

I know this is kind of stupid, but does anyone know that is there any theorem actually proved the uniqueness of eigenvector?

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What do you mean by that? – Qiaochu Yuan Jan 8 '12 at 9:28
If $\mathbf x$ is an eigenvector of $\mathbf A$, then $c\mathbf x$, when $c\neq 0$, is an eigenvector as well. – J. M. Jan 8 '12 at 9:30
@QiaochuYuan Maybe my question is not so clear. For uniqueness here I mean no other matrix can produce the same eigenvector. – Rein Jan 8 '12 at 9:35
@Rein: That's false. For example, both $\left(\begin{array}{cc}2&0\\0&3\end{array}\right)$ and $\left(\begin{array}{cc}5&0\\0&9\end{array}\right)$ have the exact same eigenspaces. And of course, any conjugate of a matrix will have the same eigenspaces with the same eigenvalues as the original matrix. So , no. Even if you require that the matrices have the same eigenspaces with the same eigenvalues and that they not be conjugate, there are examples: both of the following matrices have the same eigenspaces with the same eigenvalues (cont) – Arturo Magidin Jan 8 '12 at 9:41
(cont) $\left(\begin{array}{cccc}1 & 1 & 0 & 0\\0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1\\ 0 & 0 & 0 & 1\end{array}\right)$ and $\left(\begin{array}{cccc} 1 & 1 & 0 & 0\\ 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{array}\right)$. But they are not conjugate. – Arturo Magidin Jan 8 '12 at 9:43