What is an eigenvalue of a matrix?

For a square matrix A, a scalar λ is an eigenvalue if there is a nonzero vector v (an eigenvector) such that Av = λv. Equivalently, λ satisfies det(A − λI) = 0, where I is the identity matrix of the same size.

How do you find eigenvalues of a 2×2 matrix?

For A = [[a, b], [c, d]], eigenvalues solve λ² − tr(A)λ + det(A) = 0, where tr(A) = a + d and det(A) = ad − bc. Use the quadratic formula: λ = (tr(A) ± √(tr(A)² − 4 det(A))) / 2. If the discriminant is negative, the eigenvalues are a complex conjugate pair.

How are eigenvalues found for a 3×3 matrix?

Eigenvalues are the roots of the cubic characteristic polynomial det(A − λI) = 0. For a real matrix, this cubic always has at least one real root; the other two roots are either both real or a complex conjugate pair. This calculator finds one real root numerically, then solves the resulting quadratic factor.

What is the characteristic polynomial?

The characteristic polynomial of an n×n matrix A is p(λ) = det(A − λI). It is degree n, monic, and its roots are exactly the eigenvalues of A. The coefficient of λⁿ⁻¹ is −tr(A), and the constant term is (−1)ⁿ det(A).

What is the relationship between trace, determinant, and eigenvalues?

For any n×n matrix, the sum of eigenvalues (counting algebraic multiplicity over ℂ) equals tr(A), and the product equals det(A). For 2×2, this matches λ₁ + λ₂ = tr(A) and λ₁λ₂ = det(A).

Can a real matrix have complex eigenvalues?

Yes. Non-real eigenvalues of a real matrix always appear in complex conjugate pairs. For example, a 2×2 rotation matrix has eigenvalues on the unit circle in ℂ.

Are eigenvalues the same as eigenvectors?

No. Eigenvalues are scalars; eigenvectors are directions (nonzero vectors) that are only scaled by the matrix. Each eigenvalue has associated eigenvectors found by solving (A − λI)v = 0.

How accurate is this eigenvalue calculator?

2×2 results use the quadratic formula in double precision. 3×3 results use numerical root-finding for one real root and an exact quadratic solve for the remaining factor. Always verify critical work for exams or engineering sign-off using independent methods or software.

What are eigenvalues used for?

Eigenvalues appear in stability analysis, differential equations, principal component analysis, vibration modes, PageRank-style linear models, and diagonalization. They describe how a linear transformation scales along certain directions.

Does this calculator show eigenvectors?

This tool focuses on eigenvalues and the characteristic polynomial. For eigenvectors, use a full diagonalization workflow or solve (A − λI)v = 0 for each λ; see our related matrix tools linked below.

theCalcs

Get Custom Solution

Loading calculators...

Fetching calculator categories and tools for this section.

Go to Calculators Browse Calculators

theCalcs

Get Custom Solution

Loading the page...

Preparing tools and content for you. This usually takes a second.

Go Home Browse Calculators

theCalcs

Get Custom Solution

Linear algebra

Eigenvalue Calculator - Matrix Eigenvalue Calculator for 2×2 & 3×3

Free eigenvalue calculator for real square matrices. Enter a 2×2 or 3×3 matrix and get eigenvalues from the characteristic equation det(A − λI) = 0, plus trace, determinant, and a clear summary for homework and exam prep.

Last updated: April 13, 2026

2×2 via quadratic formula (exact formula path)

3×3 via cubic characteristic polynomial

Trace, determinant, and polynomial displayed

Need a custom linear algebra tool for your product? Get a Quote

Eigenvalue Calculator

Enter a real 2×2 or 3×3 matrix; eigenvalues update from the characteristic polynomial

Matrix size

Matrix A (row by row)

Layout: [[a,b],[c,d]]

Eigenvalues

λ1

λ2

Trace tr(A)

Determinant det(A)

Characteristic polynomial

λ² − 4λ + 3 = 0

Interpretation

Two real eigenvalues from the characteristic quadratic.

Educational note

• Eigenvalues are roots of det(A − λI) = 0.
• For 2×2: λ satisfies λ² − tr(A)λ + det(A) = 0.
• Results are numerical; verify critical work by hand or with course software.

Eigenvalue Calculator Modes & Methods

2×2 Eigenvalue Calculator

Closed-form eigenvalues from the characteristic quadratic

Characteristic equation

λ² − tr(A)λ + det(A) = 0

Ideal for first linear algebra courses and quick checks on 2×2 systems.

3×3 Eigenvalue Calculator

Roots of the cubic characteristic polynomial

Characteristic equation

λ³ − tr(A)λ² + S₂λ − det(A) = 0

S₂ is the sum of 2×2 principal minors—handled automatically from your entries.

Characteristic Polynomial Tool

See det(A − λI) encoded as a polynomial in λ

Eigenvalues = roots

p(λ) = 0

Connects determinant, trace, and eigenvalues in one view.

Real & Complex Eigenvalues

Recognize conjugate pairs from real matrices

Display format

a ± bi

When the discriminant is negative (2×2), results show complex eigenvalues.

Trace & Determinant Checks

Sum and product rules for eigenvalues

Invariants

Σλᵢ = tr(A)

Use these to sanity-check hand calculations.

Student-Friendly Workflow

Inputs update results as you type

Use cases

Homework · Labs · Review

Pair with diagonalization and determinant tools for full matrix study.

Quick example result

For A = [[2, 1], [1, 2]] (symmetric, trace 4, determinant 3):

λ₁

λ₂

Characteristic polynomial: λ² − 4λ + 3 = (λ − 1)(λ − 3).

How Our Eigenvalue Calculator Works

Eigenvalues are defined by the equation Av = λv for some nonzero vector v. Rearranging gives (A − λI)v = 0, which has a nontrivial solution only when det(A − λI) = 0. Our tool builds that determinant as a polynomial in λ and solves it: quadratics explicitly for 2×2, and a one-real-root plus quadratic factor strategy for 3×3, aligned with standard linear algebra curricula.

2×2 formula

λ = (tr(A) ± √(tr(A)² − 4 det(A))) / 2

The term under the square root is the discriminant. If it is negative, the eigenvalues are complex conjugates.

Matrix A − λI and det(A − λI) define the characteristic polynomial

Mathematical foundation

The characteristic polynomial is independent of basis: similar matrices share the same eigenvalues. Trace and determinant are coefficients of that polynomial and act as quick consistency checks after you compute λᵢ.

Eigenvalues are roots of the characteristic polynomial p(λ) = det(A − λI).
Algebraic multiplicity counts how often a root repeats; geometric multiplicity relates to nullity of A − λI.
Symmetric real matrices have real eigenvalues and an orthogonal basis of eigenvectors.
Defective matrices may lack a full basis of eigenvectors even when eigenvalues exist.

Sources & references

Linear Algebra and Its Applications — Lay, Lay, McDonaldCharacteristic polynomial, diagonalization, and eigenvalue problems
Introduction to Linear Algebra — Gilbert StrangEigenvalues, determinants, and geometric meaning of Av = λv
Khan Academy — Introduction to eigenvalues and eigenvectorsConceptual introduction with examples

Get Custom Calculator for Your Platform

Eigenvalue calculator examples

2×2 eigenvalue example

Symmetric matrix with integer eigenvalues

Matrix

A = [[2, 1], [1, 2]]
tr(A) = 4, det(A) = 3

Steps

Write λ² − 4λ + 3 = 0.
Discriminant 16 − 12 = 4.
λ = (4 ± 2) / 2 → 3 and 1.

Eigenvalues: λ₁ = 3, λ₂ = 1

Check: sum = 4 = trace, product = 3 = determinant.

Complex eigenvalue pattern (2×2)

Rotation-scaling matrices can have λ = a ± bi.

Conjugate pair shares the same real part

3×3 diagonal example

diag(1, 2, 3) has eigenvalues 1, 2, 3.

Eigenvalues appear on the diagonal

Frequently Asked Questions

Found this calculator helpful?

Share it with others studying linear algebra

Share This Calculator

Help others discover this useful tool

Copy Link

Share on Social Media

X Facebook LinkedIn WhatsApp

Share via Email

Suggested hashtags: #LinearAlgebra #Eigenvalues #Mathematics #Education #Calculator

Related Calculators

Diagonalize Matrix Calculator

Work with eigenvalues and eigenvectors for diagonalization and similarity transforms.

Use Calculator

Matrix Determinant Calculator

Compute determinants with cofactor expansion and matrix analysis.

Use Calculator

Inverse Matrix Calculator

Find the inverse of a square matrix when it exists (det ≠ 0).

Use Calculator

Gaussian Elimination Calculator

Row-reduce matrices and linear systems with elimination steps.

Use Calculator

SVD Calculator

Singular value decomposition for rectangular or square matrices.

Use Calculator

Augmented Matrix Calculator

Solve linear systems using augmented matrix operations.

Use Calculator

theCalcs

Get Custom Solution

Loading calculators...

Fetching calculator categories and tools for this section.

Go to Calculators Browse Calculators

theCalcs

Get Custom Solution

Linear algebra

Eigenvalue Calculator - Matrix Eigenvalue Calculator for 2×2 & 3×3

Last updated: April 13, 2026

2×2 via quadratic formula (exact formula path)

3×3 via cubic characteristic polynomial

Trace, determinant, and polynomial displayed

Need a custom linear algebra tool for your product? Get a Quote

Eigenvalue Calculator

Enter a real 2×2 or 3×3 matrix; eigenvalues update from the characteristic polynomial

Matrix size

Matrix A (row by row)

Layout: [[a,b],[c,d]]

Eigenvalues

λ1

λ2

Trace tr(A)

Determinant det(A)

Characteristic polynomial

λ² − 4λ + 3 = 0

Interpretation

Two real eigenvalues from the characteristic quadratic.

Educational note

• Eigenvalues are roots of det(A − λI) = 0.
• For 2×2: λ satisfies λ² − tr(A)λ + det(A) = 0.
• Results are numerical; verify critical work by hand or with course software.

Eigenvalue Calculator Modes & Methods

2×2 Eigenvalue Calculator

Closed-form eigenvalues from the characteristic quadratic

Characteristic equation

λ² − tr(A)λ + det(A) = 0

Ideal for first linear algebra courses and quick checks on 2×2 systems.

3×3 Eigenvalue Calculator

Roots of the cubic characteristic polynomial

Characteristic equation

λ³ − tr(A)λ² + S₂λ − det(A) = 0

S₂ is the sum of 2×2 principal minors—handled automatically from your entries.

Characteristic Polynomial Tool

See det(A − λI) encoded as a polynomial in λ

Eigenvalues = roots

p(λ) = 0

Connects determinant, trace, and eigenvalues in one view.

Real & Complex Eigenvalues

Recognize conjugate pairs from real matrices

Display format

a ± bi

When the discriminant is negative (2×2), results show complex eigenvalues.

Trace & Determinant Checks

Sum and product rules for eigenvalues

Invariants

Σλᵢ = tr(A)

Use these to sanity-check hand calculations.

Student-Friendly Workflow

Inputs update results as you type

Use cases

Homework · Labs · Review

Pair with diagonalization and determinant tools for full matrix study.

Quick example result

For A = [[2, 1], [1, 2]] (symmetric, trace 4, determinant 3):

λ₁

λ₂

Characteristic polynomial: λ² − 4λ + 3 = (λ − 1)(λ − 3).

How Our Eigenvalue Calculator Works

2×2 formula

λ = (tr(A) ± √(tr(A)² − 4 det(A))) / 2

The term under the square root is the discriminant. If it is negative, the eigenvalues are complex conjugates.

Matrix A − λI and det(A − λI) define the characteristic polynomial

Mathematical foundation

Eigenvalues are roots of the characteristic polynomial p(λ) = det(A − λI).
Algebraic multiplicity counts how often a root repeats; geometric multiplicity relates to nullity of A − λI.
Symmetric real matrices have real eigenvalues and an orthogonal basis of eigenvectors.
Defective matrices may lack a full basis of eigenvectors even when eigenvalues exist.

Sources & references

Linear Algebra and Its Applications — Lay, Lay, McDonaldCharacteristic polynomial, diagonalization, and eigenvalue problems
Introduction to Linear Algebra — Gilbert StrangEigenvalues, determinants, and geometric meaning of Av = λv
Khan Academy — Introduction to eigenvalues and eigenvectorsConceptual introduction with examples