For a linear operator P on a finite-dimensional space V , the property P^2= P implies
that V = the direct sum of Null P and Range P . Prove that if P^2= P , then P is
diagonalizable and its eigenvalues can only be 0 or 1.
- AndrewLv 67 年前最愛解答
The minimal polynomial for P is P^2 - P = P(P - I) = 0
The matrix P is diagonalizable because the minimal polynomial factorize into distinct linear factors.
The eigenvalues can only be 0 or 1 because these are the only roots for the minimal polynomial.
See the wikipedia page Minimal polynomial for more details on it.