Put header here and erase this reminder. Don't forget your section number!Section 0. Setup.restart; with(LinearAlgebra): with(DEtools): with(plots):Section 1. Matrix entry.A:= <<-1|2|3>,<-3|-2|-1>>;
B:= <<1,0,-1>|<3,1,-2>>;
Other requested matrices and vectorsSection 2. Matrix Operations.Operations were performed in the supplementary worksheet. The results are to be described here. Separate subsections address the five points mentioned in the project desription.Discussion(1) Addition of matrices(2) Multiplication of matrices(3) Powers of matrices(4) Scalar multiplication using star and dotSection 3. Eigenvalues, eigenvectors, and matrix exponentials3a. Eigenvectors and eigenvaluesConstruct the matrixM1:=A.B;M2:=B.A;M3:=M1^(-1);Find eigenvalues and eigenvectors(Vals1,Vecs1):=Eigenvectors(M1);(Vals2,Vecs2):=Eigenvectors(M2);(Vals3,Vecs3):=Eigenvectors(M3);Discussion(1) M1 and M2(2) M1 and M33b. Matrix Exponentials.The matrixY1:=MatrixExponential(M1,t);Checking the equationDY1:=map(diff,Y1,t); #derivative of the matrix expoentialMY1:=M1.Y1;DY1-MY1;# checks equation if zero matrixChecking initial conditionssubs(t=0,Y1); # should yield identity matrixA vector initial conditionFind solution# y1a := ???;Check that equation is satisfied.Check that initial solution is satisfied.Checking alternate formulaQ1 := <<exp(Vals1[1]*t)|0>,<0|exp(Vals1[2]*t)>>; Y1a := Vecs1.Q1.Vecs1^(-1); #alternative formula for matrix exponentialY1-Y1a; #should give zeroSection 4. Solving systems and identifying equilibriaExample M4Eigenvalues, eigenvectors, and the fundamental matrixM4:=<<1|-2>,<-2|1>>;(ValsM4,VecsM4):=Eigenvectors(M4);Y4:=MatrixExponential(M4,t);(1) Special solutionsY4a:=Y4.<0,-1>;Y4b:=Y4.<1,0>;Y4c:=Y4.<2,-1/2>;(2) Classifying the equilibriumExample M5Eigenvalues, eigenvectors, and the fundamental matrix# M5 := ???(1) Special solutions(2) Classifying the equilibriumExample M6Eigenvalues, eigenvectors, and the fundamental matrix# M6 := ????(1) Special solutions(2) Classifying the equilibriumSection 5. A nonlinear system.Part a. The equation and its critical points.F:=-(x-y)*(1-x-y); G:=x*(2+y);critpts:=solve({F,G},{x,y});dex:=diff(x(t),t)=eval(F,{x=x(t),y=y(t)});
dey:=diff(y(t),t)=eval(G,{x=x(t),y=y(t)});Part b. Trajectories.(i) Graphing some trajectoriestrange := -8..8: window:=x=-6..6,y=-6..6:inits:=[[x(0)=1,y(0)=1],[x(0)=2,y(0)=1],[x(0)=-4,y(0)=-4],
[x(0)=3,y(0)=-1],[x(0)=-1,y(0)=-1],[x(0)=1,y(0)=-1]]:DEplot([dex,dey],[x(t),y(t)],t=trange,inits,window,color=GREEN,
linecolor=[RED,BLUE,BROWN,PLUM,CORAL,BLACK],thickness=2,stepsize=0.005,
title="Trajectories: Nonlinear equation");(ii) Discussion: nature of critical points(iii) Discussion: fate of trajectoriesPart c. Linearization and stability.(i) Linearization near each equilibrium pointA := Matrix([[diff(F,x),diff(F,y)],[diff(G,x),diff(G,y)]]);A1 := eval(A,{x=0,y=0});
Eigenvalues(A1);# A2 := ????# A3 := ????# A4 := ????(ii) Discussion