For a square complex/real matrix $A$, $A$ and $A^T$ have the same set of eigenvalues, each with same algebraic multiplicities, since their characteristic polynomials are the same.
I wonder for each eigenvalue, are its geometric multiplicities for $A$ and for $A^T$ the same?
- Similar question for $A$ and $A^H$, where $H$ means conjugate and transpose, and the relation between their eigenvalues is conjugate.
Thanks!