Quick answer

λ is an eigenvalue iff det(A − λI) = 0 iff λ is a root of p(λ).

Formula

  • p(λ) = det(A − λI)
  • p(λ) = 0
  • Av = λv

Introduction

Eigenvalue problems appear as vector equations Av = λv and as polynomial equations p(λ) = 0. This guide explains why they are equivalent and how multiplicity enters. Use the Characteristic Polynomial Calculator to obtain p(λ), then study roots and eigenvectors.

Diagonalization, stability, and modal analysis in engineering courses all depend on this link.

Read this after you can expand p(λ) by hand or with the calculator.

Eigenvalue definition and roots of p(λ)

Eigenvalue definition: λ is an eigenvalue of A if there exists nonzero v with Av = λv. Rearranging gives (A − λI)v = 0, which requires A − λI to be singular.

Singularity is equivalent to det(A − λI) = 0. Therefore eigenvalues are exactly the roots of the characteristic polynomial (over an algebraically closed field such as ℂ).

Algebraic multiplicity is the exponent of (λ − λ0) in the factored p(λ). It can exceed geometric multiplicity for defective matrices.

Spectral interpretation: diagonalizable A has a basis of eigenvectors and eigenvalues on the diagonal of a similar matrix D. Factoring p(λ) is the algebraic first step toward that picture, as shown in worked polynomial examples.

Matrix diagonalization is possible when there are enough linearly independent eigenvectors. The polynomial alone does not guarantee diagonalizability.

After roots are known, set up (A − λI)v = 0 for each eigenvalue. If solving p(λ) = 0 felt difficult, review the characteristic equation setup guide for equation-solving habits.

Key equivalences

  • p(λ) = det(A − λI)
  • p(λ) = 0
  • Av = λv, v ≠ 0

Write these three lines on one index card. They prevent mixing definitions on exams.

For real symmetric matrices, eigenvalues are real even when computed through complex roots algebraically.

Step-by-step overview

  1. Compute p(λ). Use definition or calculator.
  2. Solve p(λ) = 0. List roots with algebraic multiplicity.
  3. Find eigenvectors per eigenvalue. Solve homogeneous systems (A − λI)v = 0.
  4. Check independence. Count independent eigenvectors for diagonalization questions.
  5. Interpret in context. Stability signs, modes, or scaling behavior depend on the course.

From polynomial to eigenvectors

If p(λ) = (λ − 2)²(λ + 1), eigenvalues are 2 with algebraic multiplicity 2 and −1 with multiplicity 1.

For λ = 2, solve (A − 2I)v = 0. Geometric multiplicity is the dimension of the eigenspace.

A 2×2 example with distinct eigenvalues usually has independent eigenvectors and is diagonalizable over ℂ.

Compare hand roots with calculator-produced p(λ) before investing time in eigenvector row reduction.