Eigenvalues and column space, nullspace

Is there a way to relate Eigenvalues to the column space and nullspace of a matrix?

I believe a matrices with different eigenvalues would have a different column spaces and/or nullspace. Is this correct?

I am wondering if you can prove that the Eigenvalues of $A$ and $A^T$ are equal using properties of column spaces and nullspaces.

My thinking is:
If you transform a matrix $A$ into $B$, if the row space of $B$ is orthogonal to the nullspace of $A$, and the column space of $B$ is orthogonal to the left nullspace of $A$, then matrices $A$ and $B$ have the same eigenvalues.

Solution 1:

Matrices with different eigenvalues can have the same column space and nullspace. For a simple example, consider the real 2x2 identity matrix and a 2x2 diagonal matrix with diagonals 2,3. The identity has eigenvalue 1 and the other matrix has eigenvalues 2 and 3, but they both have rank 2 and nullity 0 so their column space is all of $\mathbb{R}^2$ and their nullspace is $\{0\}$.

This is also probably a negative answer to your question about the transpose - the column space and nullspace don't contain enough information about the eigenvalues.

On the other hand, eigenvalues are certainly related to the nullspace of $A-\lambda I$, where $\lambda$ is an eigenvalue of $A$. Namely, every eigenvector must lie in the nullspace of this matrix.