Potential Diagonalizability

Example 1. The matrix ( 0 −1 1 0 ) in M2(R) is not diagonalizable, but it becomes diagonalizable in M2(C) since its characteristic polynomial splits with distinct roots in C[T ]. Example 2. The matrix ( 1 1 0 1 ) is not diagonalizable over any field. Indeed, its only eigenvalue is 1 and its only eigenvectors are scalar multiples of ( 1 0 ) , so there is never a basis of eigenvectors for this matrix.