Matrix Diagonalization Calculator
Diagonalize square matrices by finding eigenvalues and eigenvectors. Compute similarity transformations P⁻¹AP = D, analyze Jordan canonical form, and visualize eigenspaces. Get complete step-by-step solutions with geometric interpretations.
Matrix Input
Matrix Diagonalization Theory
What is Matrix Diagonalization?
Matrix diagonalization is the process of finding a diagonal matrix D that is similar to a given square matrix A. A matrix A is diagonalizable if there exists an invertible matrix P such that P⁻¹AP = D, where D is diagonal. The columns of P are eigenvectors of A, and the diagonal entries of D are the corresponding eigenvalues.
Diagonalization Formula:
A = PDP⁻¹
Where:
• P = [v₁ v₂ ... vₙ] (eigenvector matrix)
• D = diag(λ₁, λ₂, ..., λₙ) (eigenvalue matrix)
• Avᵢ = λᵢvᵢ for each eigenvalue-eigenvector pair
Diagonalizability Conditions:
• A has n linearly independent eigenvectors
• Geometric multiplicity = Algebraic multiplicity for each eigenvalue
• If A has n distinct eigenvalues, then A is diagonalizable
Diagonalization Process
Step 1: Find Characteristic Polynomial
Compute det(A - λI) to get the characteristic polynomial
Step 2: Solve for Eigenvalues
Find roots of characteristic polynomial: det(A - λI) = 0
Step 3: Find Eigenvectors
For each λᵢ, solve (A - λᵢI)v = 0 to find eigenvectors
Step 4: Check Linear Independence
Verify that eigenvectors are linearly independent
Step 5: Form Matrices P and D
P = [v₁ v₂ ... vₙ], D = diag(λ₁, λ₂, ..., λₙ)
Step 6: Verify Diagonalization
Check that AP = PD or A = PDP⁻¹
Types of Matrices
Diagonalizable Matrix
Has n linearly independent eigenvectors
Can be written as A = PDP⁻¹
Powers: Aᵏ = PDᵏP⁻¹
Non-Diagonalizable Matrix
Deficient in eigenvectors
Requires Jordan canonical form
Has Jordan blocks for repeated eigenvalues
Symmetric Matrix
Always diagonalizable
Real eigenvalues
Orthogonal eigenvectors
Normal Matrix
AA* = A*A (commutes with conjugate transpose)
Always diagonalizable
Includes symmetric, skew-symmetric, unitary matrices
Applications
Linear Systems
Solving differential equations, system dynamics, stability analysis
Principal Component Analysis
Data reduction, feature extraction, dimensionality reduction
Quantum Mechanics
Observable operators, state evolution, measurement theory
Vibration Analysis
Modal analysis, natural frequencies, mode shapes
Graph Theory
Spectral graph theory, network analysis, clustering
Machine Learning
Spectral clustering, kernel methods, manifold learning
Worked Examples
Example 1: Simple 2×2 Diagonalization
Problem:
Diagonalize the matrix:
[0 2]
Solution:
Step 1: Characteristic polynomial: det(A - λI) = (3-λ)(2-λ) = 0
Step 2: Eigenvalues: λ₁ = 3, λ₂ = 2
Step 3: Eigenvectors: v₁ = [1; 0], v₂ = [1; -1]
Step 4: P = [1 1; 0 -1], D = [3 0; 0 2]
Result: A = PDP⁻¹ with P⁻¹ = [1 1; 0 -1]
Example 2: Symmetric Matrix
Problem:
Diagonalize the symmetric matrix:
[1 2]
Solution:
Step 1: Characteristic polynomial: det(A - λI) = λ² - 4λ + 3 = 0
Step 2: Eigenvalues: λ₁ = 3, λ₂ = 1
Step 3: Eigenvectors: v₁ = [1; 1]/√2, v₂ = [1; -1]/√2
Step 4: P = [1/√2 1/√2; 1/√2 -1/√2] (orthogonal)
Result: A = PDP^T with D = [3 0; 0 1]
Example 3: Non-Diagonalizable Matrix
Problem:
Analyze the matrix:
[0 2]
Solution:
Step 1: Characteristic polynomial: det(A - λI) = (2-λ)² = 0
Step 2: Eigenvalue: λ = 2 (algebraic multiplicity 2)
Step 3: Eigenspace: dim(E₂) = 1 (geometric multiplicity 1)
Step 4: Since geometric ≠ algebraic multiplicity
Result: Matrix is NOT diagonalizable, requires Jordan form
Frequently Asked Questions
A matrix is diagonalizable if and only if it has n linearly independent eigenvectors, where n is the size of the matrix. This occurs when the geometric multiplicity equals the algebraic multiplicity for each eigenvalue.
Eigenvalues are scalars λ that satisfy Av = λv for some non-zero vector v. Eigenvectors are the non-zero vectors v that satisfy this equation. Each eigenvalue has a corresponding eigenspace of eigenvectors.
The Jordan canonical form is a block diagonal matrix that represents any square matrix when diagonalization is not possible. It consists of Jordan blocks, each corresponding to an eigenvalue, with 1's on the superdiagonal for deficient eigenvalues.
Symmetric matrices have the special property that they always have real eigenvalues and orthogonal eigenvectors. The spectral theorem guarantees that any symmetric matrix can be diagonalized by an orthogonal matrix.
The characteristic polynomial is found by computing det(A - λI), where A is your matrix, λ is the variable, and I is the identity matrix. The roots of this polynomial are the eigenvalues of the matrix.
Diagonalization is used in solving systems of differential equations, computing matrix powers efficiently, principal component analysis, quantum mechanics, vibration analysis, and many areas of engineering and physics.