# Relationship between a matrix and its eigenvectors

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?

• 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.
– Syed
Jan 11 at 19:05
• thanks for reminding me Jan 11 at 19:15
• This is best suited to the math forum.
– A.G.
Jan 11 at 23:16

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.