CALCULATE WITH CONFIDENCE

Linear Algebra & Matrices

Eigenvalue & Eigenvector Calculator

Find eigenpairs for real symmetric matrices or general real 2 × 2 matrices.

Linear Algebra & Matrices

Enter matrix values

Exact structural results; labelled numerical approximations where required
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
Try an example
Matrix A

Use signed finite decimals. Fill every visible cell. Fractions, commas, and scientific notation are not accepted.

Linear algebra result

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Enter valid values to see the result.

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Understand the subject

What are eigenvectors?

An eigenvector is a direction a matrix does not turn away from itself; the matrix only scales or flips it.

Use it to identify invariant directions of a supported transformation.

The relationship

The defining relationship

See the structure

What the calculation is doing

Worked interpretation

Read the result in context

Av=λv

Interpret with care

Important boundary

General nonsymmetric matrices can require complex eigenvalues or vectors outside this calculator’s scope.

Use the Matrix Calculator to experiment with small exact matrix operations before applying a more specialized route.

Extended domain

Complex 2 × 2 eigenpairs

A real or complex 2 × 2 matrix may have complex eigenvalues and eigenvectors. Both eigenpairs are shown.

Try an example
EIGENVALUESr1 = 1ir2 = -1i

Eigenvector directions: v₁ = [-1, 1i]ᵀ; v₂ = [-1, -1i]ᵀ (an equivalent nonzero direction may be used).

ReImr1r2

Quick guide

How to use this calculator

  1. Set the matrix dimensions required by the task.
  2. Fill every visible entry and any operation-specific control.
  3. Read the result type, factors, classification, and disclosed limitations.

Calculation method

Apply the stated linear-algebra contract

Av = λv

Undefined, singular, dimensionally incompatible, dependent, or unsupported inputs receive an explicit message instead of a fabricated numeric result.

Worked example

Example interpretation

Symmetric matrices have a real orthonormal eigenbasis.

Av = λv

Supported inputs

Precision and limits

Matrix size

Input matrices are limited to 1–6 rows and columns; operation-specific square, rank, or compatibility rules still apply.

Entry precision

Enter signed finite decimals with at most 30 digits and 15 decimal places.

Numerical policy

Results use real double-precision arithmetic, a 1e-10 structural tolerance where stated, bounded iterations, and labelled approximations.

Complex 2 × 2 extension

The established panel retains its exact or larger real-matrix workflow. The extended panel accepts complex entries for a bounded 2 × 2 analysis, reports both eigenvalues, and states branch or diagonalizability limitations.