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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.

For other matrices we use determinants and linear algebra. This is the key calculation in the chapter—almost every application starts by solving Ax = λx. First move λx to the left side. Write the equation Ax = λx as (A −λI)x = 0. The matrix A −λI times the eigenvector x is the zero vector. The eigenvectors make up the nullspace of A ...

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