Lecture Notes - Cayley-Hamilton Theorem
1. Historical Background
The Cayley-Hamilton axiom, individual of ultimate important results in undeviating
arithmetic, was first found by Arthur Cayley in 1858 while active accompanying 2×2
origins. William Rowan Hamilton later comprehensive this introduce 1864, chief to
allure current name. The axiom shows a appealing links 'tween a cast's
characteristic polynomial and allure functional characteristics.
Initially, Cayley written welcome judgments in a paper named "A Memoir on the
Theory of Matrices," that is thought-out individual of the basic everything in cast
belief. The axiom's importance wasn't directly acknowledged, but over period, it has
enhance main to miscellaneous uses in uninterrupted arithmetic, characteristic
equatings, and control belief.
2. Mathematical Prerequisites
2.1 Matrices and Determinants
Square origins: n×n arrays of numbers
Basic forge movements
Addition: (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ
Multiplication: (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ
Scalar duplication: (cA)ᵢⱼ = caᵢⱼ
Properties of cause
Inverse models
2.2 Polynomials
Definition of polynomials: p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Operations accompanying polynomials
Evaluation of polynomials
Division invention
2.3 Eigenvalues and Eigenvectors
Definition: Av = λv
Geometric understanding
Characteristic equating: det(λI - A) = 0
Algebraic and lines multiplicities
3. The Theorem: Statement and Formulation
3.1 Formal Statement
Let A be an n×n forge over a field F (usually ℝ or ℂ), and allow p(λ) = det(λI - A) be
allure characteristic polynomial. Then:
p(A) = 0
place 0 shows the n×n nothing mold.
3.2 Examples for Small Dimensions
2×2 Matrix Example
For A = [2 1; 1 2]:
Characteristic polynomial:
p(λ) = det(λI - A) = det([λ-2 -1; -1 λ-2])
= (λ-2)² - 1
= λ² - 4λ + 3
By Cayley-Hamilton:
A² - 4A + 3I = 0
3×3 Matrix Example
For A = [1 0 0; 0 2 0; 0 0 3]:
p(λ) = (λ-1)(λ-2)(λ-3)
= λ³ - 6λ² + 11λ - 6
4. Proof of the Theorem
4.1 Proof Strategy
Consider adj(λI - A), the adjugate cast
Use the correspondence (λI - A)adj(λI - A) = det(λI - A)I
Show that substituting A for λ yields p(A) = 0
4.2 Detailed Proof
Let's decay the evidence into steps:
The characteristic polynomial is:
p(λ) = det(λI - A)
By definition of adjugate mold:
(λI - A)adj(λI - A) = det(λI - A)I
Both hands are polynomial origins in λ
The equating holds for all λ
Therefore, substituting A for λ:
(AI - A)adj(AI - A) = p(A)I = 0
5. Applications and Consequences
5.1 Computing Matrix Powers
Use the axiom to express taller capacities in agreements of lower capacities
Example: If p(λ) = λ³ - 6λ² + 11λ - 6
Then A³ = 6A² - 11A + 6I
5.2 Finding Inverses
For invertible forge A:
If p(λ) = a λₙⁿ + ... + a₁λ + a₀
Then -a₀⁻¹(aₙAⁿ⁻¹ + ... + a₁I) = A⁻¹
5.3 Minimal Polynomial
The littlest polynomial divides the characteristic polynomial
Helps in judgment forge functions
Applications in resolving characteristic equatings
6. Computational Methods
6.1 Algorithm for Applying the Theorem
Calculate characteristic polynomial p(λ)
Express p(λ) in standard form
Replace λ accompanying cast A
Verify p(A) = 0
6.2 Computer Implementation
pythonCopydef characteristic_polynomial(A):
n = len(A)
λ = sympy.Symbol('λ')
return sympy.det(λ*sympy.eye(n) - A)
7. Related Theorems and Extensions
7.1 Primary Matrix Functions
If f(λ) is logical, therefore f(A) is clear
Connection to mold epidemic
Applications in characteristic equatings
7.2 Jordan Canonical Form
Connection to slightest polynomial
Decomposition of molds
Generalized eigenvectors
8. Practice Problems and Solutions
Problem 1
Find the characteristic polynomial and validate Cayley-Hamilton for:
A = [3 1; -1 2]
Solution:
p(λ) = λ² - 5λ + 7
Verify: A² - 5A + 7I = 0
Problem 2
Use Cayley-Hamilton to find A⁴ likely:
A³ - 2A² + A - I = 0
Solution:
A⁴ = 2A³ - A² + A
9. Common Pitfalls and Mistakes
9.1 Common Errors
Forgetting the similarity form term
Incorrect replacement of A
Sign wrongs in characteristic polynomial
9.2 Verification Steps
Check characteristic polynomial forecast
Verify model mathematics
Confirm things produced equals nothing mold
10. Advanced Topics and Further Study
10.1 Generalization to Other Algebras
Application to quaternions
Extension to manipulator belief
Infinite-spatial analogs
10.2 Applications administrative Theory
System establishment study
Transfer function models
State-scope likenesses
11. Historical Notes and Development
The axiom has progressed from allure original expression:
Initial finding by Cayley (1858)
Hamilton's gift (1864)
Modern proofs and generalizations
Applications in branch of quantum physics
Study Questions
State and justify the Cayley-Hamilton axiom for a 2×2 form.
How can the axiom be used to find A⁻¹?
Explain the connection betwixt the slightest and characteristic polynomials.
Apply the axiom to answer a structure of uninterrupted characteristic equatings.
Additional Resources
Linear Algebra textbooks
Online form calculators
Academic documents on requests
Interactive imagination finishes
