3.4 Cayley-Hamilton theorem

The Cayley-Hamilton theorem is a game-changer in linear algebra. It says every square matrix satisfies its own characteristic equation, linking a matrix's algebraic properties to its polynomial. This powerful tool opens doors to efficient computations and deeper insights into matrix behavior.

From finding minimal polynomials to calculating high matrix powers, this theorem's applications are far-reaching. It simplifies complex matrix operations, aids in determining diagonalizability, and even helps construct Jordan canonical forms. Understanding it is key to mastering eigenvalues and eigenvectors.

Cayley-Hamilton Theorem

Statement and Significance

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• Cayley-Hamilton theorem asserts every square matrix satisfies its own characteristic equation
• For square matrix A, characteristic polynomial p(λ) = det(λI - A) yields p(A) = 0
• Applies to matrices over any field (real numbers, complex numbers, finite fields)
• Provides polynomial equation of degree n for n × n matrix
• Connects algebraic properties of characteristic polynomial with matrix itself
• Proof involves advanced concepts (adjugate matrix, determinant properties)

Applications and Implications

• Enables expressing matrix powers as linear combinations of lower powers
• Facilitates efficient computation of high matrix powers
• Allows expressing matrix inverse as polynomial in matrix itself
• Provides insights into matrix diagonalization and Jordan canonical form
• Helps determine matrix diagonalizability without computing eigenvectors
• Aids in constructing Jordan canonical form for non-diagonalizable matrices
• Crucial for analyzing linear transformations in abstract vector spaces

Applying Cayley-Hamilton Theorem

Finding Minimal Polynomials

• Minimal polynomial defined as monic polynomial of least degree annihilating matrix A
• Cayley-Hamilton theorem guarantees minimal polynomial divides characteristic polynomial
• Process to find minimal polynomial:
  • Start with characteristic polynomial
  • Systematically test lower-degree factors
  • Compute matrix powers
  • Check linear dependencies among powers
• Minimal polynomial provides crucial information about:
  • Matrix's algebraic properties
  • Jordan canonical form
• Degree of minimal polynomial always ≤ size of matrix
• In some cases, minimal polynomial equals characteristic polynomial

Computing Matrix Powers and Inverses

• Express any matrix power as linear combination of lower powers
• For n × n matrix, An written as linear combination of I, A, A², ..., An-1
• Method particularly useful for efficiently computing high matrix powers
• Inverse of matrix expressed as polynomial in matrix itself
• For invertible matrices, provides explicit formula for A⁻¹ using powers up to An-1
• Determine coefficients by solving linear equations from characteristic polynomial
• Valuable technique when direct inversion methods computationally expensive

Matrix Powers and Inverses

Efficient Computation of Powers

• Utilize Cayley-Hamilton theorem to express high powers efficiently
• Example: For 3×3 matrix A with characteristic polynomial p(λ) = λ³ - 5λ² + 2λ - 1
  • A³ = 5A² - 2A + I
  • A⁴ = 5A³ - 2A² + A = 5(5A² - 2A + I) - 2A² + A = 23A² - 9A + 5I
• Reduces computational complexity for large powers
• Particularly useful in applications (Markov chains, graph theory)

Matrix Inverse Calculation

• Express inverse as polynomial in matrix using Cayley-Hamilton theorem
• For invertible A with characteristic polynomial p(λ) = λn + an-1λn-1 + ... + a1λ + a0
  • A⁻¹ = -(1/a0)(An-1 + an-1An-2 + ... + a2A + a1I)
• Example: 2×2 matrix A with p(λ) = λ² - 3λ + 2
  • A⁻¹ = -(1/2)(A - 3I)
• Provides alternative to traditional inverse computation methods
• Useful when dealing with symbolic matrices or in theoretical proofs

Implications for Diagonalization vs Jordan Form

Diagonalizability Criteria

• Matrix diagonalizable if and only if minimal polynomial has no repeated roots
• Cayley-Hamilton theorem aids in determining diagonalizability without eigenvector computation
• Example: Matrix with characteristic polynomial (λ - 2)²(λ - 3)
  • If minimal polynomial is (λ - 2)(λ - 3), matrix diagonalizable
  • If minimal polynomial is (λ - 2)²(λ - 3), matrix not diagonalizable
• Connects algebraic multiplicity of eigenvalues to geometric multiplicity

Jordan Canonical Form Insights

• For non-diagonalizable matrices, theorem helps construct Jordan canonical form
• Size of largest Jordan block for eigenvalue ≤ multiplicity in minimal polynomial
• Example: 4×4 matrix with minimal polynomial (λ - 2)²(λ - 3)
  • Jordan form has at most two blocks for eigenvalue 2, one block for eigenvalue 3
• Provides structural information about generalized eigenvectors
• Essential for understanding nilpotent matrices and their properties