I have a couple of questions regarding diagonalizable matrices. I think I know the answers, but I wish to confirm.
1) Is every triangular matrix also diagonalizable? I think not. The eigenvalues are the elements of the diagonal, and if one of them repeat itself, the matrix may not be diagonalizable.
2) Any matrix with distinct eigenvalues is diagonalizable. I think so. If the eigenvalues are distinct, then the algebraic multiplicity is $1$, meaning that it has to equal the geometric multiplicity.
Am I correct in both? I would appreciate it if you correct me if I am wrong with either explanation or conclusions. Thank you.