Matrix Diagonalization Calculator

Diagonalize a square matrix by computing eigenvalues, eigenvectors, and the decomposition A = PDP⁻¹. Supports 2×2 to 5×5 matrices with step-by-step solutions, characteristic polynomial, multiplicity analysis, and interactive visualization.

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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.

Eigenvalues

Scalars λ satisfying Av = λv. They appear on the diagonal of D.

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Eigenvectors

Nonzero vectors v satisfying Av = λv. They form the columns of P.

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Diagonal Matrix D

Contains eigenvalues on the diagonal, zeros elsewhere. Easy to compute powers.

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Similarity

A and D are similar matrices — they represent the same transformation in different bases.

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?

Condition	Diagonalizable?	Example
n distinct real eigenvalues	Always yes	$\begin{pmatrix}2&1\\0&3\end{pmatrix}$ → λ = 2, 3
Symmetric matrix (A = Aᵀ)	Always yes (real λ)	Spectral theorem guarantees orthogonal diagonalization
Repeated λ with AM = GM	Yes	$\begin{pmatrix}5&0\\0&5\end{pmatrix}$ → λ = 5 (AM=2, GM=2)
Repeated λ with AM > GM	No	$\begin{pmatrix}1&1\\0&1\end{pmatrix}$ → λ = 1 (AM=2, GM=1)
Complex eigenvalues	Over ℂ: check AM = GM	$\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ → λ = ±i

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

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Matrix Powers

$A^k = PD^kP^{-1}$. Since D is diagonal, $D^k$ just raises each eigenvalue to the k-th power.

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Differential Equations

Systems x' = Ax decouple into independent equations when A is diagonalized: x' = PDP⁻¹x.

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Markov Chains

Long-run behavior of transition matrices is determined by eigenvalues — diagonalization reveals steady states.

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PCA

Principal Component Analysis diagonalizes the covariance matrix to find the directions of maximum variance.

Diagonalization vs. Other Decompositions

Decomposition	Form	Requirement
Eigendecomposition (this tool)	A = PDP⁻¹	n independent eigenvectors
Spectral (symmetric)	A = QΛQᵀ	A = Aᵀ (Q orthogonal)
Jordan Normal Form	A = PJP⁻¹	Any square matrix
SVD	A = UΣVᵀ	Any matrix (even non-square)
LU Decomposition	A = LU	Square, with conditions

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.

Reference this content, page, or tool as:

"Matrix Diagonalization Calculator" at https://MiniWebtool.com/matrix-diagonalization-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-12

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Matrix Diagonalization Calculator

About Matrix Diagonalization Calculator

What Is Matrix Diagonalization?

How to Diagonalize a Matrix

When Is a Matrix Diagonalizable?

Algebraic vs. Geometric Multiplicity

Why Diagonalization Matters

Diagonalization vs. Other Decompositions

Frequently Asked Questions

Related MiniWebtools:

Linear Algebra:

Top & Updated:

Condition	Diagonalizable?	Example
n distinct real eigenvalues	Always yes	\(\begin{pmatrix}2&1\\0&3\end{pmatrix}\) → λ = 2, 3
Symmetric matrix (A = Aᵀ)	Always yes (real λ)	Spectral theorem guarantees orthogonal diagonalization
Repeated λ with AM = GM	Yes	\(\begin{pmatrix}5&0\\0&5\end{pmatrix}\) → λ = 5 (AM=2, GM=2)
Repeated λ with AM > GM	No	\(\begin{pmatrix}1&1\\0&1\end{pmatrix}\) → λ = 1 (AM=2, GM=1)
Complex eigenvalues	Over ℂ: check AM = GM	\(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) → λ = ±i