# Remark equilibrium points often have interesting

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Remark.Equilibrium points often have interesting interpretations.For example, if thelinear dynamical system describes the population dynamics of a country, with the vectorcdenoting immigration (emigration when entries ofcare negative), an equilibrium point isa population distribution that does not change, year to year. In other words, immigrationexactly cancels the changes in population distribution caused by aging, births, and deaths.
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 dierence 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?
Chapter 10Matrix multiplicationIn this chapter we introduce matrix multiplication, a generalization of matrix-vectormultiplication, and describe several interpretations and applications.10.1Matrix-matrix multiplicationIt is possible to multiply two matrices usingmatrix multiplication. You can multiplytwo matricesAandBprovided their dimensions arecompatible, which means thenumber of columns ofAequals the number of rows ofB. SupposeAandBarecompatible,e.g.,Ahas sizempandBhas sizepn. Then the product matrixC=ABis themnmatrix with elementsCij=pXk=1AikBkj=Ai1B1j+· · ·+AipBpj,i= 1, . . . , m,j= 1, . . . , n.(10.1)There are several ways to remember this rule.To find thei, jelement of theproductC=AB, you need to know theith row ofAand thejth column ofB.

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