Mathematics — Linear Algebra · diagonalizable-operators
Problem
Suppose V is a finite-dimensional complex vector space, T∈L(V), and p is the minimal polynomial of T. Prove that the following are equivalent.
((a)) T is diagonalizable.
((b)) There does not exist λ∈C such that p is a polynomial multiple of (z−λ)2.
((c)) p and its derivative p′ have no zeros in common.
((d)) The greatest common divisor of p and p′ is the constant polynomial 1.
The greatest common divisor of p and p′ is the monic polynomial q of largest degree such that p and p′ are both polynomial multiples of q. The Euclidean algorithm for polynomials (look it up) can quickly determine the greatest common divisor of two polynomials, without requiring any information about the zeros of the polynomials. Thus the equivalence of (a) and (d) above shows that we can determine whether T is diagonalizable without knowing anything about the zeros of p.