6: Spectral Theory
( \newcommand{\kernel}{\mathrm{null}\,}\)
- 6.1: Eigenvalues and Eigenvectors of a Matrix
- Spectral Theory refers to the study of eigenvalues and eigenvectors of a matrix. In this section we consider what eigenvalues and eigenvectors are and how to find them.
- 6.2: Eigenvalues and Eigenvectors for Special Matrices
- In this section we consider three kinds of matrices where we can simplify the process of finding eigenvalues and eigenvectors.
- 6.3: Diagonalization
- When a matrix is similar to a diagonal matrix, the matrix is said to be diagonalizable. We look at how to find a diagonal matrix similar to a given matrix.
- 6.4: Applications of Spectral Theory
- This section considers some applications of matrix diagonalization.
- 6.5: Markov Matrices
- In this section we look at a particular kind of matrix, called a Markov matrix, and consider some its applications.
- 6.6: Dynamical Systems
- In this section we consider discrete dynamical systems and how the techniques of linear transformations covered in previous sections can be used to find solutions.
- 6.7: Orthogonal Diagonalization
- In this section we look at matrices that have an orthonormal set of eigenvectors.
- 6.8: Singular Value Decomposition
- In this section we introduce the concept of the singular values of a matrix and consider how the matrix can be written as a special product of matrices called the singular value decomposition.
- 6.9: Special Factorizations
- In this section we explore two important factorizations of matrices: the Cholesky Factorization and the QR Factorization.
- 6.10: Quadratic Forms
- In this section we use the techniques learned in this chapter to investigate quadratic forms.