1. The problem statement, all variables and given/known data Prove or disprove: A and AT have the same eigenspaces. 2. Relevant equations 3. The attempt at a solution I know that A and AT have the same determinant and so they have the same characteristic polynomial and eigenvalues, but then if they are transposed then the stuff above and below the main diagonal is swapped then they wouldn't have the same eigenspaces? Unless they were symmetric?