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D

Matrix Diagonalization

A = PDP⁻¹ where D is diagonal (eigenvalues) and P has eigenvectors as columns. Diagonalizable iff n linearly independent eigenvectors. Powers: Aⁿ = PDⁿP⁻¹.

Concept Fundamentals
A = PDP⁻¹
Formula
diag(λ₁,…,λₙ)
D
eigenvector columns
P
PDⁿP⁻¹
Aⁿ

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Diagonalizable ⟺ n lin. indep. eigenvectors. Aⁿ = PDⁿP⁻¹ (Dⁿ trivial). Symmetric ⟹ orthogonally diagonalizable.

Key quantities
A = PDP⁻¹
Formula
Key relation
diag(λ₁,…,λₙ)
D
Key relation
eigenvector columns
P
Key relation
PDⁿP⁻¹
Aⁿ
Key relation

Ready to run the numbers?

Why: Diagonalization simplifies powers, differential equations, and Markov chains. Dⁿ is trivial (element-wise).

How: Find eigenvalues λᵢ and eigenvectors vᵢ. P = [v₁|…|vₙ], D = diag(λ₁,…,λₙ). A = PDP⁻¹.

Diagonalizable ⟺ n lin. indep. eigenvectors.Aⁿ = PDⁿP⁻¹ (Dⁿ trivial).
Sources:Gilbert Strang, Linear AlgebraWolfram MathWorld

Run the calculator when you are ready.

Diagonalize MatrixA = PDP⁻¹; D diagonal

Examples

Diagonalizable: Yes
Eigenvalues: 3.0000, 1.0000
D:
[3, 0]
[0, 1]
P:
[0.7071, -0.7071]
[0.7071, 0.7071]
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Eigenvalues

Eigenvalue Sign

Steps

[SETUP]Matrix A: [2, 1] [1, 2]
[SETUP]Goal: A = PDP⁻¹
D ext{diagonal}, P ext{eigenvectors}
[EIGENVALUES]Characteristic poly: λ² - 4λ + 3 = 0
[EIGENVALUES]Eigenvalues: λ₁ = 3.0000, λ₂ = 1.0000
[RESULT]Matrix D: [3, 0] [0, 1]
[RESULT]Matrix P: [0.7071, -0.7071] [0.7071, 0.7071]
[RESULT]A = PDP⁻¹: Verified

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

λ

D diagonal = eigenvalues

📐

P columns = eigenvectors

Key Takeaways

  • • A = PDP⁻¹: D diagonal (eigenvalues), P columns = eigenvectors.
  • • Eigenvalues: roots of det(A − λI) = 0. For 2×2: λ² − (a+d)λ + (ad−bc) = 0.
  • • Diagonalizable ⟺ n linearly independent eigenvectors.
  • • Symmetric matrices: always diagonalizable with real eigenvalues.
  • • Complex eigenvalues ⟹ not diagonalizable over R.

Did You Know?

📐Spectral Theorem: symmetric matrices have orthogonal eigenvectors and real eigenvalues.Source: Spectral
🔢A^n = PD^nP⁻¹ — powers become trivial when diagonalized.Source: Powers
⚛️Quantum mechanics: diagonalization finds energy eigenstates.Source: Physics
📊PCA: diagonalize covariance matrix to find principal components.Source: Data
🔬Jordan form generalizes diagonalization for non-diagonalizable matrices.Source: Algebra
📜Rotation matrices have complex eigenvalues (e^(±iθ)); not diagonalizable over R.Source: Geometry

How It Works

1. Characteristic polynomial

Solve det(A − λI) = 0 for eigenvalues λ.

2. Eigenvectors

For each λ, solve (A − λI)v = 0 for v.

3. Build P and D

D = diag(λ₁, λ₂, ...), P = [v₁ | v₂ | ...].

4. Verify

Check A = PDP⁻¹ (or P⁻¹AP = D).

Expert Tips

Symmetric First

Symmetric matrices are always diagonalizable. Check A = Aᵀ.

Distinct Eigenvalues

n distinct eigenvalues ⟹ diagonalizable. Repeated ⟹ check eigenspace dimension.

Rotation Matrix

[[0,-1],[1,0]] has λ = ±i. Not diagonalizable over reals.

Jordan Block

[[1,1],[0,1]] has λ=1 (double) but only 1 eigenvector — not diagonalizable.

Comparison Table

FeatureThis CalculatorNumPyManual
2×2 & 3×3✅✅✅
Step-by-step✅❌✅
Eigenvalue charts✅❌❌
Non-diag error✅✅✅
Complex λ handling✅✅⚠️

FAQ

What is diagonalization?

Writing A = PDP⁻¹ where D is diagonal. Columns of P are eigenvectors, diagonal of D are eigenvalues.

When is A diagonalizable?

When A has n linearly independent eigenvectors (n = dimension).

Why complex eigenvalues fail?

Over reals, we need real P and D. Complex λ require complex eigenvectors.

What is the characteristic polynomial?

p(λ) = det(A − λI). Its roots are the eigenvalues.

2×2 formula?

λ² − (a+d)λ + (ad−bc) = 0. Quadratic formula gives λ₁, λ₂.

Symmetric matrices?

Always diagonalizable with real eigenvalues. P can be chosen orthogonal (P⁻¹ = Pᵀ).

Jordan block?

[[1,1],[0,1]] has repeated eigenvalue 1 but only 1 eigenvector — not diagonalizable.

Uses of diagonalization?

Matrix powers A^n, solving differential equations, PCA, quantum mechanics.

Stats

2×2 formula
λ² − Tλ + D
T = trace
a + d
D = det
ad − bc
A^n
PD^nP⁻¹

Sources

  • • Strang, G. Linear Algebra and Its Applications
  • • Horn & Johnson, Matrix Analysis
  • • Khan Academy — Eigenvalues and Eigenvectors
  • • MIT 18.06 — Linear Algebra
  • • Wolfram MathWorld — Matrix Diagonalization
  • • 3Blue1Brown — Eigenvectors and Eigenvalues
Disclaimer: This calculator is for educational purposes. 3×3 eigenvalue computation uses Newton's method. For production use, prefer numerical libraries (NumPy, SciPy).
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