Economics Dynamics Problems 174

Economics Dynamics Problems 174 - 158 Economic Dynamics But...

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158 Economic Dynamics But equation (4.18) will have a nonzero solution if and only if λ is chosen such that φ ( λ ) = det( A λ I ) = 0 (4.19) The values of λ which satisfy equation (4.19) are called the eigenvalues of the matrix A , and the solution to the system that are obtained using these values are called the eigenvectors corresponding to that eigenvalue. Let us clarify these concepts with a simple example. Example 4.7 Let A = ± 11 24 ² then det( A λ I ) = ³ ³ ³ ³ 1 λ 1 λ ³ ³ ³ ³ = λ 2 5 λ + 6 = 0 Let the two roots, the two eigenvalues, of this quadratic be denoted r and s , re- spectively. Then r = 3 and s = 2 . In this example we have two distinct real roots. To determine the eigenvectors, we must substitute for a particular value of λ in the equation ( A λ i I ) v i = 0 ( i = r , s ) With λ = r = 3 then A 3 I = ± ² ± 30 03 ² = ± 21 ² whose determinant value is zero as required. Hence
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