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


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?

  • $\begingroup$ 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 $\endgroup$
    Jan 11 at 19:15
  • $\begingroup$ This is best suited to the math forum. $\endgroup$
    – 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.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.