8. Application of Eigenvalues and Eigenvectors

The eigenvalues and eigenvectors of a matrix have several applications in science, engineering, and mathematics. Eigenvalues and eigenvectors are used to solve systems of difference and differential equations, but more generally, they are used for data analysis, where the matrix \(\bf{A}\) represents data rather than coefficients of a system of equations. They introduce a simple, yet powerful relationship between a matrix and a set of special vectors and scalar values, which provides elegant solutions to some otherwise difficult problems.

Eigenvalues and eigenvectors are closely related to the Singular Value Decomposition (SVD), which has many data analysis applications and is used by MATLAB in many linear algebra functions that we used in Introduction to Linear Algebra to solve systems of equations. Eigenvalues and eigenvectors are also used in Principal Component Analysis (PCA), a data analysis tool for classification and recognition applications.

Contents

• 8.1. Introduction to Eigenvalues and Eigenvectors
• 8.2. Eigenvector Animation
• 8.3. Finding Eigenvalues and Eigenvectors
  • 8.3.1. Finding Eigenvalues
  • 8.3.2. The Power Method
  • 8.3.3. Roots of a Polynomial by Eigenvalues
  • 8.3.4. Finding Eigenvectors
• 8.4. Properties of Eigenvalues and Eigenvectors
• 8.5. Diagonalization and Powers of A
  • 8.5.1. Diagonalization
  • 8.5.2. Powers of A
• 8.6. Change of Basis and Difference Equations
  • 8.6.1. Difference Equations
  • 8.6.2. Application: Fibonacci Sequence
  • 8.6.3. Application: Markov Matrices
• 8.7. Systems of Linear ODEs
  • 8.7.1. ODE Example
  • 8.7.2. Application: Closed Loop Control Systems