PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Algebra → Linear Algebra ↓


Eigenvalues and Eigenvectors


Linear algebra is a fundamental area of mathematics, and one of its interesting concepts is eigenvalues and eigenvectors. These concepts are important for a variety of applications, ranging from solving systems of linear equations to more advanced uses in physics, computer science, and even economics. The term "eigen" comes from the German word meaning "self" or "attribute", which alludes to the concept of intrinsic properties. This document will explore these concepts extensively using clear language and examples.

Basic definitions

In short, when we talk about eigenvalues and eigenvectors, we are talking about square matrices. Let's start by breaking down these terms:

Eigenvectors

The eigenvector of a square matrix A is a non-zero vector v that changes only by a scalar factor when A is applied to it. In simple terms, eigenvectors are vectors whose direction remains unchanged when a linear transformation is applied.

Eigenvalue

The eigenvalue, often denoted by λ (lambda), is a scalar indicating the factor by which the eigenvectors are scaled during the transformation. When the matrix A is multiplied by the eigenvector v, the result is simply the eigenvector scaled by the eigenvalue:

a * v = λ * v

Calculating Eigenvalues and Eigenvectors

Finding the eigenvalues and eigenvectors of a matrix involves some calculation. Let us explain these steps with an example matrix:

Example

Consider the matrix A :

A = [2 0]
    [0 3]

Follow these steps to find the eigenvectors and eigenvalues:

Step 1: Find the Characteristic Equation

Find the characteristic equation as follows:

det(A - λI) = 0

Here, I is the identity matrix of the same size as A, and det denotes the determinant. For our matrix:

A - λI = [2-λ 0 ]
         [0 3 - λ]

Calculate the determinant:

det(A – λI) = (2-λ)(3-λ) – 0
             = λ² - 5λ + 6

Step 2: Solve the characteristic polynomial

Set the determinant to zero and solve for λ:

λ² - 5λ + 6 = 0

On factoring the quadratic equation, we get:

(λ - 2)(λ - 3) = 0

Hence λ = 2 and λ = 3 are the eigenvalues of the matrix A

Step 3: Find the eigenvectors

To find the eigenvectors, solve (A - λI)v = 0 for each λ.

When λ = 2:

[0 0] [X] [0]
[0 1] [Y] = [0]

From the above, any vector [x 0] of suitable length is an eigenvector corresponding to λ = 2.

When λ = 3:

[-1 0] [X] [0]
[0 0] [Y] = [0]

From the above, any vector [0 y] is an eigenvector corresponding to λ = 3.

Geometrical interpretation

To understand eigenvalues and eigenvectors geometrically, let us consider the effect of a matrix transformation on vectors in two-dimensional space.

Visual Example

X-axis v1 (λ=2) v2 (λ=3)

The red line represents an eigenvector (v1) where the direction does not change but the length increases by a factor of 2. The blue line represents another eigenvector (v2) where the direction remains unchanged and the length increases by a factor of 3.

Further understanding

Eigenvalues and eigenvectors have several important properties and applications:

Property

  • Determinant and trace: The determinant of a matrix is the product of its eigenvalues, and the trace (sum of diagonal elements) is the sum of its eigenvalues.
  • Invertibility: A matrix is invertible if and only if all its eigenvalues are non-zero.
  • Diagonalization: A matrix can be diagonalized if it has enough linearly independent eigenvectors.

Application

Eigenvalues and eigenvectors have a wide range of applications, including:

  • Principal Component Analysis (PCA): Used in statistics and machine learning for dimension reduction, PCA relies heavily on eigenvectors and eigenvalues.
  • Stability analysis: In differential equations and dynamical systems, eigenvalues inform us about the stability of systems.
  • Graph theory: The eigenvalues of a graph can tell a lot about its structure and properties.

Conclusion

Understanding eigenvalues and eigenvectors is foundational for anyone delving into linear algebra. Their concepts underpin many aspects of both theoretical and applied mathematics. By understanding how eigenvectors preserve direction under transformation and how eigenvalues define the scaling factor, we can delve deeper into the nuances of linear transformations and their countless applications in a variety of fields.


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