Minimal Polynomial (linear Algebra) - Applications

Applications

An endomorphism φ of a finite dimensional vector space over a field F is diagonalizable if and only if its minimal polynomial factors completely over F into distinct linear factors. The fact that there is only one factor X − λ for every eigenvalue λ means that the generalized eigenspace for λ is the same as the eigenspace for λ: every Jordan block has size 1. More generally, if φ satisfies a polynomial equation P(φ) = 0 where P factors into distinct linear factors over F, then it will be diagonalizable: its minimal polynomial is a divisor of P and therefore also factors into distinct linear factors. In particular one has:

• : finite order endomorphisms of complex vector spaces are diagonalizable. For the special case of involutions, this is even true for endomorphisms of vector spaces over any field of characteristic other than 2, since is a factorization into distinct factors over such a field. This is a part of representation theory of cyclic groups.
• : endomorphisms satisfying are called projections, and are always diagonalizable (moreover their only eigenvalues are 0 and 1).
• By contrast if with k ≥ 2 then φ (a nilpotent endomorphism) is not necessarily diagonalizable, since has a repeated root 0.

These case can also be proved directly, but the minimal polynomial gives a unified perspective and proof.