Put header here and erase this reminder. Don't forget your section number!Section 0. Setup.with(LinearAlgebra):with(DEtools):with(plots):Section 1. Matrix entry.A:= <<3|2|12>,<-4|9|14>>;
B:= <<3,4,6>|<2,3,-4>>;
C:= <<10,0>|<-2,6>|<-7,6>>;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 four 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 matrixsumsumM1:=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 M3To see why this is true: 3b. Solution of Initial Value Problems for First Order Systemsc := LinearSolve(Vecs2, <2, -1, 3>);MVals2 := <<exp(Vals2[1]*t)|0|0>, <0|exp(Vals2[2]*t)|0>, <0|0|exp(Vals2[3]*t)>>;Y1:=3c. Solution Using Matrix ExponentialsThe matrixE2:=MatrixExponential(M2,t);Checking the equationDE2:=map(diff,E2,t); #derivative of the matrix exponentialME2:=M2.E2;DE2-ME2;# checks equation if zero matrixChecking initial conditionssubs(t=0,E2); # should yield identity matrixA vector initial conditionFind solutionCheck that equation is satisfied.Check that initial solution is satisfied.Check that Y2 and Y1 are the same.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Checking alternate formulaQ2 := MVals2; #note the matrix Q2 has been computed in 3bE2a := Vecs2.Q2.Vecs2^(-1); #alternative formula for matrix exponentialE2-E2a; #should give zeroSection 4. Saddle points, nodes, and spirals.4a. Example AEigenvalues, eigenvectors, and the fundamental matrixM4A:=<<-4,2>|<-3,3>>;(Vals4A,Vecs4A):=Eigenvectors(M4A);E4A:=MatrixExponential(M4A,t);Special solutionsY4Aa:=E4A.<1,0>;Y4Ab:=E4A.<-1,0>;Y4Ac:=E4A.<1/2,-1/2>;Y4Ad:=E4A.<-1/2,1/2>;The phase planetvals:=t=-3..3;yvals:=y1=-3..3,y2=-3..3;Grapheq4A:=[diff(y1(t),t)=(M4A.<y1(t),y2(t)>)[1],diff(y2(t),t)=(M4A.<y1(t),y2(t)>)[2]];Field4A:=DEplot(eq4A,[y1(t),y2(t)],tvals,yvals,color=GREEN):Sol4A:=plot([[Y4Aa[1],Y4Aa[2],tvals],
[Y4Ab[1],Y4Ab[2],tvals],
[Y4Ac[1],Y4Ac[2],tvals],
[Y4Ad[1],Y4Ad[2],tvals]], yvals,color=[BLACK,RED,BLUE,CYAN],thickness=2):display({Field4A,Sol4A},title="Equation4A");4b. Example BEigenvalues, eigenvectors, and the fundamental matrixSpecial solutionsThe phase planeGraph4c. Example CEigenvalues, eigenvectors, and the fundamental matrixLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Special solutionsThe phase planeGraph4d. DiscussionDetermining flow direction along a trajectoryDiscussion for M4ADiscussion for M4BDiscussion for M4C4e. Saddle point graph with special trajectoriesTTdSMApJN1JUQUJMRV9TQVZFLzQzMjEwMDE2NzJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMnIiMiJCIiJCEiJSIiIyIiKiIjNyIjOUYmTTdSMApJN1JUQUJMRV9TQVZFLzQzMjEwMDIxNTJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMnIiQiIyIiJCIiJSIiJyIiI0YnISIlRiY=TTdSMApJN1JUQUJMRV9TQVZFLzQzMjEwMDI2MzJYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMnIiMiJCIjNSIiISEiIyIiJyEiKEYqRiY=