🔷 Diagonalize Matrix Calculator
Diagonalize matrix A = PDP⁻¹ using eigenvalues and eigenvectors
How to Use This Calculator
Enter Matrix
Input all elements of your 2×2 square matrix.
Calculate
Click to find eigenvalues, eigenvectors, and diagonalization.
Review Diagonalization
See P (eigenvector matrix), D (diagonal eigenvalue matrix), and verify A = PDP⁻¹.
Formula
A = P D P⁻¹
Where P contains eigenvectors, D is diagonal with eigenvalues
Diagonalization Process:
- Find eigenvalues λᵢ by solving det(A - λI) = 0
- Find eigenvectors vᵢ for each eigenvalue: (A - λᵢI)vᵢ = 0
- Form P = [v₁ | v₂ | ... | vₙ] (eigenvectors as columns)
- Form D = diag(λ₁, λ₂, ..., λₙ) (eigenvalues on diagonal)
- Then A = P D P⁻¹
Requirements:
- Matrix must be square
- Must have n linearly independent eigenvectors
- Eigenvalues can be real or complex
When is a matrix diagonalizable?
- If it has n distinct eigenvalues (sufficient condition)
- If it has n linearly independent eigenvectors (necessary and sufficient)
- Symmetric matrices are always diagonalizable
About Diagonalize Matrix Calculator
The Diagonalize Matrix Calculator finds the diagonalization A = PDP⁻¹ of a square matrix. The diagonal matrix D contains eigenvalues, and the matrix P contains corresponding eigenvectors as columns. Diagonalization simplifies many matrix operations.
When to Use This Calculator
- Linear Algebra: Understand matrix structure
- Matrix Powers: Compute Aⁿ = P Dⁿ P⁻¹ easily
- Differential Equations: Solve systems of linear ODEs
- Dynamical Systems: Analyze stability and behavior
- Signal Processing: Principal component analysis
Why Use Our Calculator?
- ✅ Complete Solution: Shows eigenvalues, eigenvectors, P, and D
- ✅ Step-by-Step: Clear diagonalization process
- ✅ Verification: Demonstrates A = PDP⁻¹ structure
- ✅ Educational: Helps understand diagonalization
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Diagonalizable: A matrix can be written as A = PDP⁻¹
- Eigenvalues: Scalars λ such that Av = λv
- Eigenvectors: Non-zero vectors v such that Av = λv
- P Matrix: Columns are eigenvectors (must be invertible)
- D Matrix: Diagonal matrix with eigenvalues
Applications
Matrix Powers: Aⁿ = P Dⁿ P⁻¹, where Dⁿ is just eigenvalues raised to power n.
Exponential: eᴬ = P eᴰ P⁻¹ for matrix exponentials.
Frequently Asked Questions
What does it mean to diagonalize a matrix?
Diagonalizing a matrix A means finding matrices P and D such that A = PDP⁻¹, where D is diagonal. This simplifies many matrix operations.
Can every matrix be diagonalized?
No, only matrices with n linearly independent eigenvectors are diagonalizable. Some matrices (defective matrices) cannot be diagonalized, but can be put in Jordan form.
What if eigenvalues are complex?
For real matrices, complex eigenvalues come in conjugate pairs. Diagonalization is still possible using complex numbers, or you can use real Jordan form.
Are symmetric matrices always diagonalizable?
Yes! Real symmetric matrices are always diagonalizable with real eigenvalues and orthogonal eigenvectors. This is the Spectral Theorem.
Why is diagonalization useful?
Diagonalization simplifies matrix operations: Aⁿ = P Dⁿ P⁻¹, det(A) = product of eigenvalues, trace(A) = sum of eigenvalues, and matrix functions become easier.