Eigenvalues and eigenvectors are key concepts in matrix theory. They help us understand how matrices transform space, revealing hidden structures and behaviors. This knowledge is crucial for solving complex problems in physics, engineering, and data analysis.

In this section, we'll dive into the math behind eigenvalues and eigenvectors. We'll learn how to calculate them, explore their geometric meaning, and uncover important properties that make them so useful in real-world applications.

Eigenvalues and eigenvectors of matrices

Fundamental concepts

• Eigenvalues (λ) represent scalar values that scale vectors when a matrix is applied to them
• Eigenvectors (v) constitute non-zero vectors that remain in the same direction when multiplied by a matrix
• Eigenvalue equation expresses the relationship as $Av = λv$, where A represents a square matrix
• Square matrices of size n × n have at most n distinct eigenvalues (real or complex numbers)
• Eigenvalues and eigenvectors apply exclusively to square matrices due to their relation to linear transformations
• Eigenspace refers to the set of all eigenvectors corresponding to a particular eigenvalue

Mathematical properties

• Trace of a matrix equals the sum of its eigenvalues
• Determinant of a matrix equals the product of its eigenvalues
• Similar matrices share identical eigenvalues but typically have different eigenvectors
• For any integer k, if λ represents an eigenvalue of A, then λ^k represents an eigenvalue of A^k
• Triangular matrices have eigenvalues located on their main diagonal
• Real symmetric matrices possess real eigenvalues and orthogonal eigenvectors for distinct eigenvalues
• Eigenbasis forms when normalized eigenvectors of a diagonalizable matrix are arranged as columns

Calculating eigenvalues and eigenvectors

Characteristic equation method

• Solve the characteristic equation $det(A - λI) = 0$ to find eigenvalues, where I represents the identity matrix
• Characteristic polynomial roots yield the matrix eigenvalues
• For each eigenvalue λ, solve the homogeneous system $(A - λI)v = 0$ to determine corresponding eigenvectors
• Algebraic multiplicity refers to an eigenvalue's multiplicity as a characteristic polynomial root
• Geometric multiplicity denotes the dimension of an eigenvalue's corresponding eigenspace
• Explicit formulas calculate eigenvalues for 2x2 and 3x3 matrices (quadratic and cubic equations)
• Numerical methods often required for larger matrices (power method, QR algorithm)

Special cases and simplifications

• Triangular matrices reveal eigenvalues directly on their main diagonal
• Diagonal matrices have eigenvalues equal to their diagonal entries
• Symmetric matrices guarantee real eigenvalues and orthogonal eigenvectors
• Unitary matrices have eigenvalues with magnitude 1
• Nilpotent matrices possess only 0 as an eigenvalue
• Companion matrices have characteristic polynomials equal to their defining polynomials

Geometric interpretation of eigenvalues and eigenvectors

Linear transformations

• Eigenvectors represent directions in which linear transformations scale, rotate, or reflect vectors
• Positive real eigenvalues indicate stretching or compression along the eigenvector direction (λ > 1 stretching, 0 < λ < 1 compression)
• Negative real eigenvalues represent reflection and scaling along the eigenvector direction
• Complex eigenvalues correspond to rotations combined with scaling in the plane spanned by eigenvector components
• Eigenvalue of 1 signifies unchanged vectors in the corresponding eigenvector direction
• Determinant represents the factor by which the transformation changes volumes (product of eigenvalues)

Visualization in different dimensions

• In 2D: Eigenvectors define invariant lines under the transformation
  • Real distinct eigenvalues: Two invariant lines (stretch/compress along each)
  • Complex conjugate eigenvalues: Rotation and scaling in the plane
  • Repeated eigenvalue: One invariant line, possible shear in perpendicular direction
• In 3D: Eigenvectors with real eigenvalues define principal axes of the transformation
  • Three real distinct eigenvalues: Three invariant lines
  • One real and two complex conjugate eigenvalues: One invariant line and a plane of rotation
  • Three repeated eigenvalues: Possible scaling in all directions or more complex behavior

Properties of eigenvalues and eigenvectors

Algebraic relationships

• Sum of eigenvalues equals the trace of the matrix $\sum_{i=1}^n λ_i = tr(A)$
• Product of eigenvalues equals the determinant of the matrix $\prod_{i=1}^n λ_i = det(A)$
• Eigenvalues of matrix powers: If λ represents an eigenvalue of A, then λ^k represents an eigenvalue of A^k
• Eigenvalues of matrix inverses: If λ ≠ 0 represents an eigenvalue of A, then 1/λ represents an eigenvalue of A^(-1)
• Eigenvalues of matrix polynomials: If λ represents an eigenvalue of A, then p(λ) represents an eigenvalue of p(A) for any polynomial p

Structural properties

• Similar matrices (P^(-1)AP) share identical eigenvalues but generally different eigenvectors
• Transpose matrices (A^T) have the same eigenvalues as the original matrix A
• Complex conjugate matrices have complex conjugate eigenvalues
• Hermitian matrices (A^H = A) have real eigenvalues and orthogonal eigenvectors
• Normal matrices (AA^H = A^HA) have orthogonal eigenvectors
• Positive definite matrices have all positive real eigenvalues