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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.

Chapter 6 Eigenvalues and Eigenvectors 6.1 Introduction to Eigenvalues 1 An eigenvector x lies along the same line as Ax : Ax = λx. The eigenvalue is λ. 2 If Ax = λx then A2x = λ2x and A−1x = λ−1x and (A + cI)x = (λ + c)x: the same x. 3 If Ax = λxthen (A−λI)x = 0andA−λI is singularand det(A−λI) = 0. neigenvalues.

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