Matrix Diagonalization Calculator

About Matrix Diagonalization Calculator

The Matrix Diagonalization Calculator decomposes any square matrix into the form A = PDP⁻¹, where D is a diagonal matrix of eigenvalues and P is the matrix of eigenvectors. Enter a 2×2 to 5×5 matrix and get the complete factorization with step-by-step solutions, characteristic polynomial, algebraic and geometric multiplicity analysis, and interactive animation of the decomposition.

What Is Matrix Diagonalization?

Matrix diagonalization is the process of finding matrices P and D such that:

$$A = PDP^{-1}$$

where D is a diagonal matrix whose entries are the eigenvalues of A, and P is an invertible matrix whose columns are the corresponding eigenvectors. Equivalently, \(D = P^{-1}AP\), which means D is similar to A.

How to Diagonalize a Matrix

Step 1. Select the matrix size (2×2 to 5×5) and enter values into the grid. You can also click a quick example to load a preset matrix for testing.

Step 2. Click Diagonalize Matrix. The calculator computes the characteristic polynomial det(A − λI) and finds its roots (eigenvalues).

Step 3. For each eigenvalue, the tool solves (A − λI)x = 0 to find eigenvectors, and checks algebraic vs. geometric multiplicity to determine if the matrix is diagonalizable.

Step 4. If diagonalizable, the calculator constructs P (eigenvectors as columns), D (eigenvalues on diagonal), and P⁻¹, then verifies PDP⁻¹ = A.

Step 5. Explore the animated decomposition to visualize how A factors into P × D × P⁻¹, and step through the full solution using the navigation controls.

When Is a Matrix Diagonalizable?

Algebraic vs. Geometric Multiplicity

For each eigenvalue λ:

• Algebraic multiplicity (AM): the number of times λ appears as a root of the characteristic polynomial det(A − λI) = 0.

• Geometric multiplicity (GM): the dimension of the eigenspace ker(A − λI), i.e., the number of linearly independent eigenvectors.

A matrix is diagonalizable if and only if GM = AM for every eigenvalue. The condition 1 ≤ GM ≤ AM always holds.

Why Diagonalization Matters

Diagonalization vs. Other Decompositions

Frequently Asked Questions

What does it mean to diagonalize a matrix?

Diagonalizing a matrix A means finding an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹. The diagonal entries of D are the eigenvalues, and the columns of P are the corresponding eigenvectors.

When is a matrix diagonalizable?

A matrix is diagonalizable if and only if, for every eigenvalue, the geometric multiplicity equals the algebraic multiplicity. Equivalently, there must be n linearly independent eigenvectors for an n×n matrix. All symmetric real matrices and all matrices with n distinct eigenvalues are diagonalizable.

What is the difference between algebraic and geometric multiplicity?

The algebraic multiplicity is how many times an eigenvalue appears as a root of the characteristic polynomial. The geometric multiplicity is the dimension of the eigenspace, i.e., the number of linearly independent eigenvectors for that eigenvalue. A matrix is diagonalizable precisely when these two quantities are equal for every eigenvalue.

Can a matrix with complex eigenvalues be diagonalized?

Yes, a matrix with complex eigenvalues can still be diagonalized over the complex numbers, as long as the geometric multiplicity equals the algebraic multiplicity for each eigenvalue. The resulting P and D matrices will contain complex entries.

What are the applications of matrix diagonalization?

Matrix diagonalization is used to compute matrix powers efficiently (A^k = PD^kP⁻¹), solve systems of differential equations, analyze Markov chains and steady-state behavior, perform principal component analysis in statistics, and understand linear transformations in physics and engineering.