**Terminology:**- eigenvalue, eigenvector, characteristic polynomial, eigenspace, similar matrices, similarity transformation, diagonalization.
**Objectives:**- learn how to determine eigenvalues and eigenvectors for a matrix, learn how to diagonalize a matrix.
**Reading Assignment:**- Chapter 5, Sections 5.1-5.3 (pages 295-319).
**Lesson Outline****Key Ideas and Discussion:**-
Eigenvalues and eigenvectors are needed for the analysis of many types of
mathematical models. In these problems, there is an equation of the form
, where
*A*is an matrix. A solution to this problem consists of an eigenvalue , and an associated eigenvector . Usually there are*n*eigenvalue-eigenvector solutions to . The problem of finding the eigenvalues and eigenvectors for a square real matrix*A*is a very different kind of problem compared to the problems that have been solved so far in this course, where the primary tool has been elementary row operations. Finding eigenvalues is equivalent to finding roots of a special polynomial (the characteristic polynomial). This degree*n*polynomial can easily be determined for small matrices using a special determinant, . When evaluating the determinant to find , a determinant expansion formula should be used. You might be tempted to try to use elementary row operations, but these can not easily be used because the diagonal elements in the determinant depend on . Even though the characteristic polynomial will have real coefficients, the roots (eigenvalues) can in some cases be complex numbers. There is one type of matrix where eigenvalues can easily be found; if a matrix is triangular (it has all entries zero below the main diagonal, or all entries zero above the main diagonal), then the eigenvalues are the diagonal entries for the matrix. Once an eigvenvalue has been found, an associated eigenvector can be found by solving the appropriate homogeneous system, . Notice that if is an eigenvector, then so is for any scalar*s*; eigenvectors are unique only up to scalar multiples. The eigenspace for a particular eigenvalue is the same as . This usually has dimension equal to one; but sometimes the dimension can be larger (for example, if is a repeated eigenvalue).If the matrix

*A*is transformed, using a nonsingular matrix*P*, into another matrix*B*with , then*A*and*B*are said to be similar. Similar matrices have the same eigenvalues. The aim of diagonalization is to find a matrix*P*so that*B*is a diagonal matrix*D*. In this case, the eigenvalues of*A*(and*D*) are the diagonal entries of*D*. Diagonalization is possible if and only if*A*has*n*linearly independent eigenvectors, which can be used for the columns of*P*, so that*AP*=*PD*. **Practice Problems:**- 5.1.1, 5, 11, 13, 21 (pp. 302-303);
5.2.3, 11, 17, 21 (pp. 311-312);
5.3.1, 5, 11, 15, 21 (pp. 319-320).
**Assignment**-

Thu May 28 14:15:52 PDT 1998