Eigenvalues and eigenvectors are key to understanding linear transformations. They help us break down complex operations into simpler parts, making it easier to analyze and solve problems in linear algebra.

Characteristic polynomials are tools for finding eigenvalues, while eigenspaces show us how eigenvectors behave. These concepts are crucial for diagonalization, which simplifies matrix operations and has wide-ranging applications in science and engineering.

Characteristic Polynomials of Matrices

Definition and Properties

• Characteristic polynomial of a square matrix A defined as $det(λI - A)$
  • λ represents a variable
  • I denotes the identity matrix
  • det signifies the determinant
• For linear operator T on finite-dimensional vector space V, characteristic polynomial expressed as $det(λI - [T])$
  • [T] represents matrix representation of T relative to chosen basis
• Degree of characteristic polynomial equals dimension of vector space or size of square matrix
• Characteristic polynomial remains unchanged regardless of basis chosen for linear operator representation
• Roots of characteristic polynomial correspond to eigenvalues of matrix or linear operator
• Characteristic polynomial factorization takes form $p(λ) = (λ - λ₁)^m₁ (λ - λ₂)^m₂ ... (λ - λₖ)^mₖ$
  • λᵢ represent distinct eigenvalues
  • mᵢ denote their algebraic multiplicities

Applications and Examples

• Characteristic polynomials used to determine eigenvalues and eigenvectors
• Example: For 2x2 matrix A = [[3, 1], [1, 3]], characteristic polynomial calculated as: $det(λI - A) = det([[λ-3, -1], [-1, λ-3]]) = (λ-3)² - 1 = λ² - 6λ + 8$
• Characteristic polynomials aid in analyzing matrix properties (determinant, trace)
• Applications in various fields (physics, engineering, computer graphics)

Finding Eigenvalues

Solving the Characteristic Equation

• Characteristic equation obtained by setting characteristic polynomial to zero: $det(λI - A) = 0$
• Solution methods for characteristic equation include:
  • Factoring (for simpler polynomials)
  • Quadratic formula (for second-degree polynomials)
  • Advanced techniques (for higher-degree polynomials)
• 2x2 and 3x3 matrices often solvable by hand
• Larger matrices may require computational methods (numerical algorithms)
• Complex eigenvalues occur in conjugate pairs for real matrices

Properties and Special Cases

• Sum of eigenvalues (counting multiplicity) equals trace of matrix
• Product of eigenvalues equals determinant of matrix
• Eigenvalues of triangular matrix (diagonal matrices) found on main diagonal
• Example: For upper triangular matrix A = [[2, 1, 3], [0, 4, -2], [0, 0, 1]], eigenvalues are 2, 4, and 1
• Special matrices (symmetric, orthogonal) have specific eigenvalue properties
  • Symmetric matrices have real eigenvalues
  • Orthogonal matrices have eigenvalues with magnitude 1

Eigenvalue Multiplicities

Algebraic and Geometric Multiplicities

• Algebraic multiplicity of eigenvalue λ defined as power of (λ - λᵢ) in factored characteristic polynomial
• Geometric multiplicity of eigenvalue λ equals dimension of corresponding eigenspace
  • Calculated as nullity of (A - λI)
• Geometric multiplicity always less than or equal to algebraic multiplicity
• Simple eigenvalue defined as having algebraic multiplicity of 1
• Defective eigenvalue has geometric multiplicity strictly less than algebraic multiplicity
• Sum of all algebraic multiplicities equals dimension of vector space or size of square matrix

Examples and Applications

• Example: Matrix A = [[2, 1, 0], [0, 2, 0], [0, 0, 3]] has characteristic polynomial $(λ-2)²(λ-3)$
  • Eigenvalue 2 has algebraic multiplicity 2, geometric multiplicity 1 (defective)
  • Eigenvalue 3 has algebraic and geometric multiplicity 1 (simple)
• Multiplicities crucial for determining matrix diagonalizability
• Applications in stability analysis of dynamical systems

Eigenspaces of Matrices

Finding and Representing Eigenspaces

• Eigenspace E_λ for eigenvalue λ defined as set of all eigenvectors associated with λ, including zero vector
• To find eigenspace, solve homogeneous system $(A - λI)v = 0$
  • v represents an eigenvector
• Eigenspace E_λ equivalent to null space of matrix (A - λI)
• Basis vectors for eigenspace found by:
  1. Reducing (A - λI) to row echelon form
  2. Solving for free variables
• Dimension of eigenspace E_λ equals geometric multiplicity of λ
• Eigenvectors corresponding to distinct eigenvalues are linearly independent

Examples and Applications

• Example: For matrix A = [[3, 1], [1, 3]] with eigenvalues 2 and 4:
  • Eigenspace for λ = 2: E₂ = span{[1, -1]}
  • Eigenspace for λ = 4: E₄ = span{[1, 1]}
• Eigenspaces used in modal analysis of vibrating systems
• Applications in quantum mechanics (energy eigenstates)

Properties of Eigenspaces

Fundamental Characteristics

• Eigenspaces function as subspaces of vector space on which linear operator acts
• Sum of dimensions of all eigenspaces less than or equal to dimension of vector space
• Operator deemed diagonalizable if sum of eigenspace dimensions equals vector space dimension
• Eigenspaces corresponding to distinct eigenvalues:
  • Form linearly independent subspaces
  • Have trivial intersection (only zero vector)
• Direct sum of all eigenspaces creates invariant subspace under linear operator
• When operator has n distinct eigenvalues (n = space dimension), corresponding eigenvectors form basis for entire space

Applications and Examples

• Example: 3x3 rotation matrix around z-axis has eigenspaces:
  • E₁ = span{[0, 0, 1]} (real eigenvalue)
  • Complex eigenspaces for e^(iθ) and e^(-iθ)
• Eigenspace decomposition used in:
  • Principal Component Analysis (data analysis)
  • Solving systems of differential equations
• Understanding eigenspace properties crucial for:
  • Analyzing matrix powers
  • Exponentiating matrices