This week we take up matrix/vector methods for solving higher-order differential equations and systems of differential equations. To some extent, we will be retracing some of the ground already covered -- e.g., second-order linear equations with constant coefficients. However, we will be using those examples to illustrate powerful tools for understanding more general equations -- and it is useful to have examples for which we already know the answers.
The central problem of Chapter 7 is to solve a first-order homogeneous system of linear differential equations in the form X' = AX, where X is a vector of unknown functions, and A is the matrix of (constant) coefficients. We will see that the key to finding solutions is to first find nonzero vectors v such that Av = rv, where r is a number. That is, we want to find the direction vectors v for which the effect of multiplication by A is to produce a vector in the same (or opposite) direction. A number r for which this is possible is called an eigenvalue for A, and any corresponding nonzero vector v is called an eigenvector for r (or for A).
The prefix eigen is the German word for characteristic. The primary reason for using a German name rather than an English one is that it is less than half as long -- and therefore takes less time and space for writing it, which we will have to do a lot.
The defining equation for eigenvalues and eigenvectors can also be written in the form
(A - rI) v = 0.
Thus, the "eigenproblem" is to find nonzero solutions of a homogeneous system of linear equations with the same number of rows as columns. Such solutions exist only if the matrix of coefficients, A-rI, is singular -- in which case there must be infinitely many solutions. So we first need to ask what numbers can be subtracted from each diagonal entry of A to produce a singular matrix -- and how to find those numbers. The problem of finding eigenvalues and eigenvectors will keep us busy all week. In lab we will see how to use these concepts in Maple, and next week we will apply the results to differential equations.
Here is the syllabus for Week 9:
Week 9 | Date | Topic | Reading | Activity |
M | 10/26 | Eigenvalues and eigenvectors |
7.3b | |
W | 10/28 | Eigenvalues and eigenvectors |
7.3b | |
F | 10/30 | Eigenvalues and eigenvectors |
7.3b | Lab: Eigenvalues and Eigenvectors |
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Last modified: July 14, 1998