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Chapter 6 Eigenvalues and Eigenvectors

Chapter 6 Eigenvalues and introduction to Eigenvalues '&$%1 Aneigenvectorxlies along the same line asAx:Ax= .2 IfAx= xthenA2x= 2xandA 1x= 1xand(A+cI)x= ( +c)x: the xthen(A I)x=0andA Iis singular anddet(A I)= s bydetA= ( 1)( 2) ( n)and diagonal suma11+a22+ +ann=sum of have =1and0. Reflections have1and 1. Rotations haveei ande i :complex!This Chapter enters a new part of linear algebra. The first part was aboutAx=b:balance and equilibrium and steady state. Now the second part is aboutchange. Timeenters the picture continuous time in a differential equationdu/dt=Auor time stepsin a difference equationuk+1=Auk. Those equations are NOT solved by key idea is to avoid all the complications presented by the matrixA. Supposethe solution vectoru(t)stays in the direction of a fixed vectorx.

6.1. Introduction to Eigenvalues 289 To explain eigenvalues, we first explain eigenvectors. Almo st all vectors change di-rection, when they are multiplied by A. Certain exceptional vectors x are in the same direction as Ax. Those are the “eigenvectors” . Multiply an eigenvector by A, and the vector Ax is a number λ times the original x.

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