Linear Algebra II
Mathematics 271
Course
Structure
Unit 4
Eigenvalues and Eigenvectors
Introduction
Unit 4 goes further into the concepts of “eigenvalue,” “eigenvector,” and “eigenspace,” and demonstrates techniques for finding real eigenvalues and their corresponding eigenspaces for n × n matrices. This discussion provides a basis for determining the conditions under which it is possible, by an appropriate change of basis, to transform a matrix into a diagonal matrix.
Objectives
After completing this unit, you should be able to
- define the terms “eigenvalue,” “eigenvector” and “eigenspace.”
- find the eigenvalues, eigenvectors, and bases for the eigenspaces of square matrices.
- solve problems on linear operators in two and three dimensions involving real and distinct eigenvalues, real but not distinct eigenvalues, and imaginary eigenvalues.
- discuss the implications of the properties of a matrix for the properties of the corresponding eigenspaces, and for the ease with which the action of the matrix on a vector can be visualized.
- determine whether or not a given square matrix is diagonalizable, and if so, find the matrix that diagonalizes it.
- determine whether or not a given square matrix is orthogonally diagonalizable, and if so, find the orthogonal matrix that diagonalizes it.
- explain when a general n × n matrix will have n linearly independent orthogonal eigenvectors, and when the eigenvectors of such a matrix will be linearly independent but not orthogonal.
