158Economic DynamicsBut equation (4.18) will have a nonzero solution if and only ifλis chosen suchthatφ(λ)=det(A−λI)=0(4.19)The values ofλwhich satisfy equation (4.19) are called theeigenvaluesof thematrixA, and the solution to the system that are obtained using these valuesare called theeigenvectorscorresponding to that eigenvalue. Let us clarify theseconcepts with a simple example.Example 4.7LetA=±11−24²thendet(A−λI)=³³³³1−λ1−−λ³³³³=λ2−5λ+6=0Let the two roots, the two eigenvalues, of this quadratic be denotedrands, re-spectively. Thenr=3 ands=2.In this example we have two distinct real roots.To determine the eigenvectors, we must substitute for a particular value ofλinthe equation(A−λiI)vi=0(i=r,s)Withλ=r=3 thenA−3I=±−²−±3003²=±−21−²whose determinant value is zero as required. Hence
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