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Given a matrix $A$ and its eigenvalue $λ$ and its eigenvector $\vec{x}$.

Problem:

From the definition that $A\vec{x}=λ\vec{x}$, can I say that eigenvectors $\vec{x}$ must be all in $A$'s column space and thus say that the rank of matrix $A$ must be not less than the number of $A$'s independent eigenvectors?

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  • $\begingroup$ Welcome to the Mathematica Stack Exchange. This stack site is about the technical computing software called Mathematica and the associated Wolfram Language. I believe your question can be more suitably answered at the Mathematics Stack Exchange. I also note that you haven't asked a specific question yet. $\endgroup$
    – Syed
    Jan 11 at 19:05
  • $\begingroup$ thanks for reminding me $\endgroup$
    – BABYFONGGG
    Jan 11 at 19:15
  • $\begingroup$ This is best suited to the math forum. $\endgroup$
    – A.G.
    Jan 11 at 23:16
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Almost correct! From $A\vec{x}=λ\vec{x}$ we get $A(\frac1\lambda\vec{x})=\vec{x}$ and thus $\vec x$ is in the column space of $A$ ... unless $\lambda=0$, in which case this argument doesn't work. (Consider $A$ equal to the zero matrix, for example.) However, you can probably recover a valid implication if you consider only eigenvectors with nonzero eigenvalues.

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  • $\begingroup$ got it ! thanks a lot!!! $\endgroup$
    – BABYFONGGG
    Jan 12 at 11:09

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