Exercises1759.4Reducing a Markov model to a linear dynamical system.Consider the 2-Markov modelxt+1=A1xt+A2xt-1,t= 2,3, . . . ,wherextis ann-vector. Definezt= (xt, xt-1). Show thatztsatisfies the linear dynamicalsystem equationzt+1=Bzt, fort= 2,3, . . ., whereBis a (2n)⇥(2n) matrix. This ideacan be used to express anyK-Markov model as a linear dynamical system, with state(xt, . . . , xt-K+1).9.5Fibonacci sequence.The Fibonacci sequencey0, y1, y2, . . .starts withy0= 0,y1= 1, andfort= 2,3, . . .,ytis the sum of the previous two entries,i.e.,yt-1+yt-2.(Fibonacciis the name used by the 13th century mathematician Leonardo of Pisa.)Express thisas a time-invariant linear dynamical system with statext= (yt, yt-1) and outputyt,fort= 1,2, . . .. Use your linear dynamical system to simulate (compute) the Fibonaccisequence up tot= 20. Also simulate a modified Fibonacci sequencez0, z1, z2, . . ., whichstarts with the same valuesz0= 0 andz1= 1, but fort= 2,3, . . .,ztis the di↵erence ofthe two previous values,i.e.,zt-1-zt-2.9.6Recursive averaging.Suppose thatu1, u2, . . .is a sequence ofn-vectors. Letx1= 0, andfort= 2,3, . . ., letxtbe the average ofu1, . . . , ut-1,i.e.,xt= (u1+· · ·+ut-1)/(t-1).Express this as a linear dynamical system with input,i.e.,xt+1=Atxt+Btut,t= 1,2, . . .(with initial statex1= 0).Remark.This can be used to compute the average of anextremely large collection of vectors, by accessing them one-by-one.9.7Complexity of linear dynamical system simulation.Consider the time-invariant lineardynamical system withn-vector statextandm-vector inputut, and dynamicsxt+1=Axt+But,t= 1,2, . . .. You are given the matricesAandB, the initial statex1, and theinputsu1, . . . , uT-1. What is the complexity of carrying out a simulation,i.e., computingx2, x3, . . . , xT?About how long would it take to carry out a simulation withn= 15,m= 5, andT= 105, using a 1 Gflop/s computer?