If A is a 2x2 matrix with rank 1, and one eigenvalue is known to be nonzero, then is A diagonalizable? I think this matrix is diagonalizable because if you take a generic 2x2 matrix with rank one:
[ a b ]
and you find the characteristic polynomial, (a-t)(bx-t) - bax = 0, then one eigenvalue must be zero and thus A is diagonalizable. Is this reasoning correct? Thanks!