20-01-2016, 03:47 PM
Introduction to Eigenvalues
Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution of du=dt D Au is changing with time growing or decaying or oscillating. We can’t find it by elimination. This chapter enters a new part of linear algebra, based on Ax D x. All matrices in this chapter are square. A good model comes from the powers A; A2; A3;::: of a matrix. Suppose you need the hundredth power A100. The starting matrix A becomes unrecognizable after a few steps, and A100 is very close to Œ :6 :6I :4 :4 : :8 :3 :2 :7 :70 :45 :30 :55 :650 :525 :350 :475 :6000 :6000 :4000 :4000
A A2 A3 A100
A100 was found by using the eigenvalues of A, not by multiplying 100 matrices. Those eigenvalues (here they are 1 and 1=2) are a new way to see into the heart of a matrix. To explain eigenvalues, we first explain eigenvectors. Almost all vectors change direction, when they are multiplied by A. Certain exceptional vectors x are in the same direction as Ax. Those are the “eigenvectors”.