09.03.2016
Dear Student,
I hope that you are enjoying the Labor Day weekend! Just in case you should have the urge to look ahead to the next week’s work in Math 488, I am sending you this weekly message on Saturday rather than the the usual Sunday time:
Our class schedule for Week 3 prescribes that you should complete Lectures 7, 8, 9 and the associated homework by Sunday, Sept. 11. I hope that the comments and suggestions below will help you as you work through this material.
Week 3: Content Summary for Lectures 7, 8 and 9:
After a brief review of Lecture 6, Lecture 7 begins the development of the eigenvalue-eigenvector approach to the solution of first-order systems of linear differential equations, a development that will continue through Lecture 11:
When the matrix A of an n by n homogeneous system y’(t) = A y(t) is a matrix with n linearly independent eigenvectors (that is, when A is a diagonalizable) the solution method is very direct and straight forward: One can write down a complete set of solutions of the given system even when some of the eigenvalues of A are complex.
We show in Lecture 7 that if lambda is a eigenvalue of A and if v(lambda) is a corresponding eigenvector, then e^(lambda(t)) v(lambda) is a solution of the system. Therefore, if A has n linearly independent eigenvectors, we can write down a complete set of linearly independent solutions of the given system of differential equations,
This approach is illustrated with several worked-out examples and Just Do It! as well as in the homework problems Lecture 7 and 8. Be sure to go over these examples carefully to see how this approach works.
Lecture 9 continues the work begun in Lectures 7 and 8 of finding a complete set of solutions for a homogeneous system of linear differential equations y’(t) = A y(t) when A is a matrix that does not have n linearly independent eigenvectors. For this purpose, we introduce and develop the so-called exponential matrix e^(A t) and develop some of its basic algebraic and analytic properties.
To motivate the definition of the exponential matrix e^(A t) of the system, we begin by reviewing some from undergraduate calculus about ordinary exponential f(t) = e^(a t):
1) e^(t) can be expanded in a power series e^(at) = 1 + a t + [ (a t)^2 / 2!] +……+ [(at^n] / n!] +….
2) This power series converges for all real t to e^(a t).
It turns out that the corresponding power series of matrices also converges term by term and the resulting matrix can be used to compute a complete set y1(t), y2(t) ,……, yn(t) of any homogeneous system y’(t) = e^(A t) where v is any vector!
If lambda is an eigenvalue of A and v(lambda) is a corresponding eigenvector,then [e^(A t)v reduces to a single term e^(lambda t) v, which is the same as the solution obtained in Lecture 7.
There is much for you to learn this week but it is important and useful stuff.
Please contact me by e-mail or my cell phone if you have any questions.
Have a productive and Interesting week!
Tony Peressini
Cell Ph. 217-840-2871
anthonyperessini@gmail.com