Quick answer

Characteristic p(λ) = det(A − λI) has degree n. Minimal m(λ) is monic of least degree with m(A) = 0.

Formula

  • p(λ) = det(A − λI)
  • m(A) = 0
  • same eigenvalues

Introduction

Introductory courses emphasize p(λ) because it is computable from determinants. Advanced matrix theory introduces the minimal polynomial as a sharper annihilating polynomial. Use the Characteristic Polynomial Calculator for p(λ); minimal polynomials are usually found by theory, not this tool.

Learn characteristic polynomials thoroughly first.

This article compares concepts so you do not confuse degrees, factors, or eigenvalue statements.

Key differences and shared eigenvalues

Characteristic polynomial: defined by det(A − λI), always degree n, always monic up to the (−1)n convention your class uses for det(A − λI).

Minimal polynomial: monic polynomial of least degree such that m(A) = 0 when λ is replaced by A using matrix arithmetic.

Both share the same eigenvalues over ℂ, but minimal polynomial factors may use smaller exponents for those eigenvalues.

Cayley-Hamilton states p(A) = 0, so the characteristic polynomial annihilates A. The Cayley-Hamilton guide walks through matrix substitution.

Educational example: if A = cI, minimal polynomial may be λ − c while characteristic is (λ − c)n.

For computing p(λ) in homework, follow the determinant formula guide rather than minimal polynomial algorithms.

Comparison snapshot

  • p(λ) = det(A − λI), degree n
  • m(λ) minimal, degree ≤ n
  • Eigenvalues: same set

Minimal polynomial divides the characteristic polynomial in the polynomial ring framework used in advanced courses.

Defective matrices highlight differences between algebraic and geometric multiplicity beyond this article's scope.

Step-by-step overview

  1. Compute p(λ) first. Use determinant definition or calculator.
  2. List eigenvalues from p(λ). Roots with multiplicity from p.
  3. Study minimal polynomial separately. Use course theorems, not the home calculator.
  4. Compare degrees. Minimal degree can be strictly smaller.
  5. Apply in Jordan form context. When your syllabus reaches advanced topics.

Side-by-side illustration

For a diagonal matrix with distinct eigenvalues λ1, …, λn, both polynomials often align up to monic scaling conventions.

For nilpotent blocks, minimal polynomial may be λk while characteristic has higher degree factors.

Always compute p(λ) for standard homework unless the problem names minimal polynomial explicitly.