8.8 Iterated Matrix Models In In jut Ayn transition matrix Givengo what is Ja Tn A Y see 8 5 Fact0If Ais a diagonal matrix a it call tl then A 2 2 nutria FactIIfAhas two distinct eigenvalues with eigenvector VT VT thenthe matrix Pi is invertible eg a 1 5with ti i 122 withv3 Y let P I d P1.41 31.17 0 É is alsoinvertible A itself may notbe inverted

Yo Fact2 letIt bea 22 matrixwith two distinct eigenvaluesddeandlet Pbe the matrix i.e ht D Then I diagonal matrix APPD Aloisi 11 Sine Pis invertible aiAI it in the exists p APP PDP AT AT A PDP Avidate Diagonals of A A in o re it1lie

An PDP YIPDP.tlDI1 Pl PDPTPDP'PSDAP EP DP PILED PD p gATo PDpgo AsCoyas I whatis P yo Ahas fur distinct eigenvalues Suppose pig E JnPD E pp't P E P I I X Pli p tic taxi till a c Civ Cat in I c I's.ttis

ExampleConsideracertain omnivorous specious suppose through observation veg meat SupposeonDay 0 If Yoo Inthe long run how many individuals will becontainment on eachday A T I Pll 31 hiA q let's eigenvalues eigenveton uses of the matrix En det g a 4 1 3 d5 Iit f f N421 42 0 I

121211I 10afat 11 1 0 d 12 121 Eigenvectors For Mi Tz Ide th E 1 11 1 II H Itsy oya 1 Not unique tty 1 in I H For X l I g 11,411 1

1 1 X I Kia o 1 E In ftp.T Alternatelycandecompose gointoa comb of EE 1gat t cat c f c g 30 4I4 C 100 C y g c GC o 2 10 In n Ciri a c i h Co f i o y ti o 10.1414

Uploaded by ConstableProtonHawk249 on coursehero.com