Transcription of Chapter 6 Eigenvalues and Eigenvectors
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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. Then we only need tofind the number (changing with time) that multipliesx.
and never get mixed. The eigenvectors of A100 are the same x 1 and x 2. The eigenvalues of A 100are 1 = 1 and (1 2) 100 = very small number. Other vectors do change direction. But all other vectors are combinations of the two eigenvectors. The first column of A is the combination x 1 +(.2)x 2: Separate into eigenvectors Then multiply by A .8.2 ...
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