8. Application of Eigenvalues and Eigenvectors

Eigenvalue / Eigenvector problems are one of the more important linear algebra topics. Eigenvalues and eigenvectors are used to solve systems of differential equations, but more generally they are used for data analysis, where the matrix represents data rather than coefficients of a system of equations. They introduce a simple, yet very powerful relationship between a matrix and a set of special vectors and scalar values. This simple relationship 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 used in principal component analysis, which is an important data analysis tool for data classification and recognition applications.

• 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. Roots of a Polynomial by Eigenvalues
  • 8.3.3. 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