A study buddy and I were going through this question and while we made some progress, we were ultimately unsure if what we were doing was correct or not.
The question is:
Let A be a diagonalizable matrix and let X be the diagonalizing matrix. Show that the column vectors of X correspond to the nonzero eigenvalues of R(A).
What we did was this:
Say that A is n x n. Therefore X must be n x n and a matrix D must be n x n since if A is diagonalizable and X is the diagonalizing matrix:
A = X D X $^{-1}$
Since X is invertible, its eigenvectors must be linearly independent. Therefore, because X is n x n, there are n linearly independent eigenvectors. These eigenvectors span $\mathbb{R}$$^n$.
Because of this, R(A) $\subset$ $\mathbb{R}$$^n$, we've shown that the the column vectors of X correspond to the nonzero eigenvalues of R(A)... except we haven't. My buddy and I got stuck here and we weren't sure what to do since we haven't accounted for the eigenvalues that are equal to 0.
Thanks for any help in advance.
Hint: Let $\lambda_1,\dots\lambda_n$ be the eigenvalues of $A$ with associated eigenvectors $\mathbf v_1,\dots,\mathbf v_n$. Collect the equations $A\mathbf v_i=\lambda_i\mathbf v_i$ into a single equation.