The Minimal Polynomial and Some Applications

The easiest matrices to compute with are the diagonal ones. The sum and product of diagonal matrices can be computed componentwise along the main diagonal, and taking powers of a diagonal matrix is simple too. All the complications of matrix operations are gone when working only with diagonal matrices. If a matrix A is not diagonal but can be conjugated to a diagonal matrix, say D := PAP−1 is diagonal, then A = P−1DP so Ak = P−1DkP for all integers k, which reduces us to computations with a diagonal matrix. In many applications of linear algebra (e.g., dynamical systems, differential equations, Markov chains, recursive sequences) powers of a matrix are crucial to understanding the situation, so the relevance of knowing when we can conjugate a nondiagonal matrix into a diagonal matrix is clear. We want look at the coordinate-free formulation of the idea of a diagonal matrix, which will be called a diagonalizable operator. There is a special polynomial, the minimal polynomial (generally not equal to the characteristic polynomial), which will tell us exactly when a linear operator is diagonalizable. The minimal polynomial will also give us information about nilpotent operators (those having a power equal to O). All linear operators under discussion are understood to be acting on nonzero finitedimensional vector spaces over a given field F .