Diagonalize Matrix Calculator
Find eigenvalues, eigenvectors, and diagonalize matrices using the formula A = PDP⁻¹
Matrix Input
Diagonalization Results
Eigenvalues
Eigenvectors
Matrix P
Eigenvectors as columns
Matrix D
Eigenvalues on diagonal
Matrix P⁻¹
Inverse of P
Diagonalization formula: A = PDP⁻¹
Verification: The original matrix A can be reconstructed by multiplying P × D × P⁻¹
Step-by-Step Diagonalization
Step 1: Find Eigenvalues
Solve the characteristic equation det(A - λI) = 0
Eigenvalues: 0.000, 0.000
Step 2: Find Eigenvectors
For each eigenvalue λᵢ, solve (A - λᵢI)v = 0
v₍1₎ = [1.000, 0.000]
v₍2₎ = [1.000, 0.000]
Step 3: Form Matrices P and D
P = [v₁ | v₂ | ...] (eigenvectors as columns), D = diagonal matrix of eigenvalues
Matrix P:
[ 1.000, 1.000] [ 0.000, 0.000]
Matrix D:
[ 0.000, 0.000] [ 0.000, 0.000]
Step 4: Verify A = PDP⁻¹
The original matrix A can be reconstructed using the diagonalization formula
Example Matrices
2×2 Diagonal Example
[3 0]
[0 1]
2×2 Symmetric Example
[1 2]
[2 1]
3×3 Example
[1 0 0]
[2 1 -1]
[0 -1 1]
Key Properties
Diagonal Matrix Benefits
- • Easy to compute powers: A^n = PD^nP⁻¹
- • Simple matrix functions
- • Efficient computations
Diagonalizability Conditions
- • Must have n linearly independent eigenvectors
- • Each eigenvalue's geometric = algebraic multiplicity
- • Symmetric matrices are always diagonalizable
Quick Tips
Eigenvalues appear on the diagonal of matrix D
Eigenvectors form the columns of matrix P
Real symmetric matrices always have real eigenvalues
Use diagonalization for computing high matrix powers
Understanding Matrix Diagonalization
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. This transformation simplifies many matrix operations and reveals important properties of the original matrix.
Applications
- •Computing high powers of matrices efficiently
- •Solving systems of differential equations
- •Principal Component Analysis (PCA)
- •Quantum mechanics and eigenstate analysis
The Diagonalization Process
- 1. Find eigenvalues by solving det(A - λI) = 0
- 2. For each eigenvalue, find corresponding eigenvectors
- 3. Check if there are enough linearly independent eigenvectors
- 4. Form matrix P using eigenvectors as columns
- 5. Form diagonal matrix D using eigenvalues
- 6. Verify that A = PDP⁻¹
When is a Matrix Diagonalizable?
• An n×n matrix is diagonalizable if it has n linearly independent eigenvectors
• Matrices with distinct eigenvalues are always diagonalizable
• Symmetric matrices are always diagonalizable over real numbers
• Some matrices with repeated eigenvalues may not be diagonalizable