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Eigenvalues and Eigenvectors - Massachusetts Institute of ...

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. Thoseeigenvalues (here they are1and1=2) are a new way to see into the heart of a explain Eigenvalues , we first explain Eigenvectors .

Rx D x. Now we use determinants and linear algebra. This is the key calculation in the chapter—almost every application starts by solving Ax D x. First move x to the left side. Write the equation Ax D x as .A I/ x D 0. The matrix A I times the eigenvector x is the zero vector. The eigenvectors make up the nullspace of A I . When we know an ...

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