Diagonalize Matrix Calculator

Diagonalize a 2x2 or 3x3 square matrix. This tool finds the eigenvalues and eigenvectors of your matrix A, then builds the eigenvector matrix P and diagonal matrix D so that A = PDP^-1.

Enter your matrix entries below. If the matrix does not have enough independent real eigenvectors, the calculator will tell you it cannot be diagonalized over the reals.

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Method verified Standard linear algebra: characteristic polynomial, eigenvalues and eigenvectors, A = PDP^-1.

1. Matrix Size

Size of square matrix A

2. Matrix Entries

Matrix A (row by row)

Diagonalization Result

Diagonalizable?
-
Enter a matrix

Eigenvalues

Roots of characteristic polynomial

Determinant of A

Product of eigenvalues

Matrix P (eigenvectors as columns)

Matrix D (eigenvalues on diagonal)

Working: A = P D P^-1

Summary: Enter your matrix to diagonalize it.

How to Diagonalize a Matrix

Diagonalizing a square matrix A means writing it as A = PDP^-1, where D is a diagonal matrix holding the eigenvalues of A, and P is a matrix whose columns are the corresponding eigenvectors. This is useful because diagonal matrices are far easier to work with, particularly when raising a matrix to a power or solving systems of linear equations.

Steps to Diagonalize

Find the characteristic polynomial. Solve det(A - λI) = 0 for λ. For a 2x2 matrix this gives a quadratic; for a 3x3 matrix it gives a cubic.
Solve for the eigenvalues. The roots λ₁, λ₂, ... of the characteristic polynomial are the eigenvalues of A.
Find the eigenvectors. For each eigenvalue λ, solve (A - λI)v = 0 to find a non-zero eigenvector v.
Build P and D. P has the eigenvectors as its columns, in the same order as the eigenvalues are placed on the diagonal of D.
Check A = PDP^-1. This only works if P is invertible, which requires n linearly independent eigenvectors for an n by n matrix.

When a Matrix Cannot Be Diagonalized

Not every square matrix can be diagonalized using real numbers. A matrix fails to diagonalize over the reals when its eigenvalues are complex, or when a repeated eigenvalue does not have enough independent eigenvectors to span its full multiplicity (a defective matrix). In these cases you would need complex eigenvalues or a more general Jordan normal form instead of a true diagonal matrix.

Worked Example

Take the 2x2 matrix A = [[2, 1], [1, 2]]. The characteristic equation is (2 - λ)² - 1 = 0, giving eigenvalues λ₁ = 3 and λ₂ = 1. Solving (A - 3I)v = 0 gives eigenvector [1, 1], and (A - I)v = 0 gives eigenvector [1, -1]. So P = [[1, 1], [1, -1]] and D = [[3, 0], [0, 1]], and A = PDP^-1.

Frequently Asked Questions

What does it mean to diagonalize a matrix? Diagonalizing a square matrix A means finding a diagonal matrix D and an invertible matrix P such that A = PDP^-1. The columns of P are the eigenvectors of A, and the entries of D are the corresponding eigenvalues in the same order.

Why do some matrices fail to diagonalize? A matrix fails to diagonalize over the reals when it does not have enough linearly independent eigenvectors, or when its eigenvalues are complex, such as a matrix with a repeated eigenvalue but only one independent eigenvector.

What is diagonalization used for? It simplifies repeated matrix operations. Once A = PDP^-1, A raised to the power k becomes PD^kP^-1, which is used in differential equations, Markov chains and dynamical systems.

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Matrix Determinant Calculator: work out the determinant.
Matrix Rank Calculator: check the rank of a matrix.

Sources: Standard linear algebra methods for eigenvalues, eigenvectors and matrix diagonalization, as presented in undergraduate linear algebra texts (for example Gilbert Strang, Introduction to Linear Algebra).

This calculator handles real eigenvalues for 2x2 and 3x3 matrices. Results are rounded for display. Matrices with complex eigenvalues or repeated eigenvalues with insufficient independent eigenvectors cannot be diagonalized over the real numbers, and this will be shown in the result.