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Let $A$ be a real square matrix, and let its column space be $$\mathrm{col}(A)=\{y\in\mathbb{C}^n:y=Ax\text{ for some } x\in\mathbb{C}^n\}.$$

  1. Under what conditions is $\mathrm{col}(A)$ spanned by eigenvectors of $A$ associated to nonzero eigenvalues?
  2. Do we get the same answer if we take $\mathbb{R}$ rather than $\mathbb{C}$ as the field (so $\mathrm{col}(A)=\{y\in\mathbb{R}^n:y=Ax\text{ for some } x\in\mathbb{R}^n\}$ and we look for conditions for the real and imaginary parts of the eigenvectors of $A$ associated to nonzero eigenvalues to be in $\mathrm{col}(A)$?

My attempt to answer 1 so far: Any eigenvector $v\in\mathbb{C}^n$ is in $\mathrm{col}(A)$, so $\mathrm{col}(A)$ is spanned by eigenvectors of $A$ associated to nonzero eigenvalues if there are $\mathrm{rank}(A)$ linearly independent eigenvectors eigenvectors of $A$ associated to nonzero eigenvalues. Can we say more? As for 2, I'm confused.

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