Eigenvalue & Eigenvector Calculator with Steps

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Formula at a glance

Eigenvalue Calculator: Characteristic equation — det(λI − A) = 0. Example: Symmetric 2 × 2 → Eigenvalues: 1, 3 — How the Eigenvalue Calculator works: Characteristic equation — det(λI − A) = 0

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📘 About the Eigenvalue & Eigenvector Calculator

An eigenvector of a matrix is a direction that the matrix only stretches, and its eigenvalue is the stretch factor: Av = λv. Eigenvalues describe vibration frequencies, stability of systems, Google’s PageRank, principal component analysis and quantum states.

Enter a square matrix. The calculator builds the characteristic polynomial det(λI − A), finds all its roots (real and complex, exact for 2 × 2 and rational cases), and gives a basis of eigenvectors for each real eigenvalue, with checks that the trace and determinant equal the sum and product of the eigenvalues.

🧮 Formula

Characteristic equation

det(λI − A) = 0

Eigenvector

(A − λI)v = 0

2 × 2

λ² − (trace)λ + det = 0

✏️ Worked examples

Symmetric 2 × 2

You enter
Matrix A	2 1; 1 2

You get
Eigenvalues	1, 3
Characteristic polynomial	det(λI − A) = λ² − 4λ + 3
Trace = sum of eigenvalues	4
Determinant = product of eigenvalues	3
Symmetric matrix	All eigenvalues are real and the eigenvectors are orthogonal

Eigenvalues 1, 3.

3 × 3 matrix

You enter
Matrix A	2 0 0; 0 3 4; 0 4 9

You get
Eigenvalues	1, 2, 11
Characteristic polynomial	det(λI − A) = λ³ − 14λ² + 35λ − 22
Trace = sum of eigenvalues	14
Determinant = product of eigenvalues	22
Symmetric matrix	All eigenvalues are real and the eigenvectors are orthogonal

Eigenvalues 1, 2, 11.

🧭 How to use this calculator

What you enter

Matrix ARows separated by ; or new lines, entries by spaces or commas. Fractions like 1/2 are fine.

What you get

Eigenvalues
Characteristic polynomial
Trace = sum of eigenvalues
Determinant = product of eigenvalues
Symmetric matrix

❓ Frequently asked questions

What is an eigenvalue in simple terms?

A number λ for which some non-zero vector v satisfies Av = λv — the matrix just scales v by λ.

Can eigenvalues be complex?

Yes. A rotation matrix such as [0 −1; 1 0] has eigenvalues ±i and no real eigenvectors.

Are eigenvectors unique?

No — any non-zero multiple of an eigenvector is also an eigenvector, so they are given up to scale.

λ Eigenvalue & Eigenvector Calculator