Economics Dynamics Problems 225

Economics Dynamics Problems 225 - Discrete systems of...

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Discrete systems of equations 209 For λ = s = 3wehave ±² 21 12 ³ ² 30 03 ³´± v s 1 v s 2 ´ = ± 0 0 ´ i.e. ± 11 1 1 ´± v s 1 v s 2 ´ = ± 0 0 ´ Hence, v s 1 + v s 2 = 0. Let v s 1 = 1, then v s 2 = v s 1 = 1. Thus, the second eigenvector is v s = ± 1 1 ´ Our matrix, V , is therefore V = µ v r v s = ± ´ From the theorem we have D = V 1 AV , i.e. V 1 = ± ´ 1 ± ´ = ± 10 ´ which is indeed the matrix D formed from the characteristic roots of A . Since D = V 1 then VDV 1 = V ( V 1 ) V 1 = A Furthermore A 2 = ( VDV 1 )( VDV 1 ) = VD 2 V 1 A 3 = ( VDV 1 )( VD 2 V 1 ) = VD 3 V 1 . . . A t = ( VDV 1 )( VD t 1 V 1 ) = VD t V 1 Hence u t = A t u 0 = VD t V 1 u 0 (5.5) or u t = V ± r t 0 0 s t ´ V 1 u 0 We can summarise the procedure as follows: (1) Given a ±rst-order linear homogeneous equation system u t = Au
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