Minimal Polynomial of a Matrix

Compare the smallest monic polynomial m(x) that annihilates a square matrix A (so that m(A) = 0) with its characteristic polynomial p(x) = det(xI - A) as eigenvalues and Jordan blocks coalesce.

Experiment

Observe how eigenvalue coalescence and superdiagonal 1s change the minimal degree in https://riano.app/module/minimal-polynomial:

1. Select Matrix family Diagonal with Lambda = 1 and Mu = 2, and verify that Minimal degree is 2 with ascending coefficients 2, -3, 1.
2. Change Lambda to 2 so that λ = μ = 2. Watch the minimal degree drop from 2 to 1 with m(x) = x - 2 (ascending coefficients -2, 1), while p(x) becomes -8 + 12x - 6x² + x³.
3. Switch to Jordan 2 plus scalar with Lambda = -2 and Mu = -2 to see that the 2x2 Jordan block forces minimal degree 2 with m(x) = (x + 2)² = 4 + 4x + x².