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.