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Linear Algebra & Matrices

Matrix Diagonalization Calculator

Find a real numerical eigenbasis P and diagonal D when supported.

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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Understand the subject

What is diagonalization?

Diagonalization changes to an eigenvector basis, scales each coordinate independently, then changes back.

Use it when a supported matrix has enough independent eigenvectors.

The relationship

The defining relationship

See the structure

What the calculation is doing

Worked interpretation

Read the result in context

A=PDP⁻¹

Interpret with care

Important boundary

Not every matrix is diagonalizable, and this tool has a bounded real-valued scope.

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

Extended domain

Complex 2 × 2 diagonalization

Distinct complex eigenvalues provide a complex eigenbasis even when a real diagonalization does not exist.

Try an example
EIGENVALUESr1 = 1ir2 = -1i

D = diag(1i, -1i); P uses the corresponding eigenvector columns.

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

A = PDP⁻¹

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

Worked example

Example interpretation

A matrix is diagonalizable only when it has enough independent eigenvectors.

A = PDP⁻¹

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.