MAT 17B(H) Lecture 25

8.8 Iterated Matrix Models In In jut Ayn transition matrix Given go what is Ja Tn A Y see 8 5 Fact 0 If A is a diagonal matrix a it call tl then A 2 2 nutria Fact I If A has two distinct eigenvalues with eigenvector VT VT then the matrix P i is invertible e g a 1 5 with t i i 12 2 with v3 Y let P I d P 1.4 1 31.1 7 0 É is also invertible A itself may not be inverted

Yo Fact 2 let It be a 2 2 matrix with two distinct eigenvalues d de and let P be the matrix i.e ht D Then I diagonal matrix AP PD Aloisi 11 Sine Pis invertible ai AI it in the exists p APP PDP AT AT A PDP Avi date Diagonals of A A in o re it 1 lie

An PDP YIPDP.tl DI1 Pl PD PTP D P'PSD AP EP DP PILED PD p g A To PD p go As Coy as I what is P yo A has fur distinct eigenvalues Suppose pig E Jn PD E pp't P E P I I X Pli p tic taxi till a c Civ Cat in I c I's.tt is

Example Consider a certain omnivorous specious suppose through observation veg meat Suppose on Day 0 If Yoo In the long run how many individuals will be containment on each day A T I Pll 31 hi A q let's eigenvalues eigenveton uses of the matrix E n det g a 4 1 3 d 5 I it f f N 421 42 0 I

1212 11 I 1 0 a fat 1 1 1 0 d 12 12 1 Eigenvectors For Mi Tz Ide th E 1 11 1 II H Itsy o ya 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 Alternately can decompose go into a comb of E E 1 g at t cat c f c g 30 4 I 4 C 100 C y g c G C o 2 10 In n Ciri a c i h C o f i o y ti o 10.14 14