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Eigenvalues and Eigenvectors

Chapter 6 Eigenvalues and Introduction to EigenvaluesLinear equationsAxDbcome from steady state problems. Eigenvalues have their greatestimportance indynamic problems. The solution ofdu=dtDAuis changing with time growing or decaying or oscillating. We can t find it by elimination. This chapter enters anew part of linear algebra, based onAxD x. All matrices in this chapter are good model comes from the powersA; A2;A3;:::of a matrix. Suppose you need thehundredth powerA100. The starting matrixAbecomes unrecognizable after a few steps,andA100is very close to :6 :6I:4 :4 : :8 :3:2 :7 :70 :45:30 :55 :650 :525:350 :475 :6000 :6000:4000 :4000 AA2A3A100A100was found by using theeigenvaluesofA, not by multiplying 100 matrices .

For those vectors, Px1 D x1 (steady state) and Px2 D 0 (nullspace). This example illustrates Markov matrices and singular matrices and (most important) symmetric matrices. All have special ’s and x’s: 1. Each column of P D:5 :5:5 :5 adds to 1,so D 1 is an eigenvalue. 2. P is singular,so D 0 is an eigenvalue. 3.

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