Characteristic and Minimal Polynomials: Notes and FAQs

The characteristic polynomial of a square matrix A is det(A minus lambda I), a degree n polynomial whose roots are exactly the eigenvalues. It packs the whole eigenvalue story, multiplicities included, into one object you can factor with ordinary polynomial tools.

The minimal polynomial goes deeper: the smallest-degree monic polynomial m with m(A) = 0. It records exactly how each eigenvalue behaves inside A, trimming repetitions the characteristic polynomial keeps, which is why it decides diagonalisability and controls matrix functions like the exponential in differential equations.

Key Concepts

• Characteristic polynomial chi_A(lambda) = det(A minus lambda I): degree n, roots are eigenvalues with algebraic multiplicities.
• Minimal polynomial m_A: unique monic polynomial of least degree with m_A(A) = 0.
• Cayley-Hamilton: chi_A(A) = 0, so m_A always divides chi_A.
• Same roots: both polynomials share the same distinct roots; only multiplicities can differ.

Formulas

• Cayley-Hamilton: chi_A(A) equals the zero matrix.
• Diagonalisability test: A diagonalises exactly when m_A splits into distinct linear factors, no repeats.
• Finding m_A: among divisors of chi_A with the same distinct roots, the smallest-degree one annihilating A wins.

Worked Example

For A = [[2,1],[0,2]]: chi_A = (2 minus lambda) squared = lambda squared minus 4 lambda + 4. Check Cayley-Hamilton: A squared = [[4,4],[0,4]], so A squared minus 4A + 4I = [[0,0],[0,0]]. For m_A, candidates are (lambda minus 2) and its square. But A minus 2I = [[0,1],[0,0]], not zero, so the minimal polynomial is (lambda minus 2) squared, equal to chi_A here. The repeated factor means A is not diagonalisable, matching the 1D eigenspace for eigenvalue 2.

Common Misconceptions

• Minimal always equals characteristic. For the identity, chi = (lambda minus 1) to the n but m = lambda minus 1, since I minus I = 0 already.
• Repeated roots in chi mean not diagonalisable. Only repeats in the minimal polynomial prove that; the identity has fully repeated chi yet is diagonal.

Real-World Applications

• ODE systems: m_A determines solution forms of x' = Ax and the matrix exponential.
• Control theory: the minimal polynomial governs controllability and feedback design.
• Graph theory: the characteristic polynomial of an adjacency matrix encodes network structure.
• Numerical methods: Krylov solvers implicitly build the minimal polynomial to solve huge linear systems without forming inverses.

The characteristic polynomial lists every eigenvalue with full multiplicity; the minimal polynomial trims each factor to what A genuinely needs. Test divisors from smallest degree upward, and read diagonalisability off the factorisation.