HW5.m

%% Problem 1
clc, clear, close all;

As = cell(3,1);
As{1} = [2,3;3,1];
tht = pi/8;
As{2} = [cos(tht),-sin(tht);sin(tht),cos(tht)];
As{3} = [1,-1;2,-1];

figure();
hold on;
colors = ['r' 'g' 'b' 'c' 'm' 'y'];
nvects = length(colors);
vects = zeros(2,nvects);
for i=1:nvects
    tht = 2*pi*(i-1)/(nvects);
    vects(:,i) = [cos(tht);sin(tht)];
end
for mat=1:3
    subplot(1,3,mat);
    hold on;
    title(['Plot of A ' num2str(mat)])
    A = As{mat};
    for i=1:nvects
        quiver(0,0,vects(1,i),vects(2,i),colors(i),'LineWidth',2);
    end
    Av = A*vects;
    for i=1:nvects
        quiver(0,0,Av(1,i),Av(2,i),[':' colors(i)],'LineWidth',2);
    end
    [eigv,eigV]=eigs(As{mat})
end


%% Problem 3
clc, clear;
As = cell(3,1);
As{1} = [8,-8,-2;
        4,-3,-2;
        3,-4,1];
As{2} = [1,0,-4;
    0,3,0;
    -2,0,-1];
As{3} = [2,1,1;
    0,3,1;
    0,-1,1];
for mat = 1:3
    disp("Problem 3 Matrix " + mat)
    [geigv,eigV] = eigs(As{mat})
    [Jordan] = jordan(As{mat})
    
end

%% Problem 6
clc, clear;
A = [2,1;1,-8];

% a.
a = zeros(2);
for i=0:4
    a = a + A^i / factorial(i);
end
a

b = zeros(2);
for i=0:4
    b = b + (-1)^i * A^i / factorial(i);
end
b = b^-1;
b

l1 = -3 - sqrt(36+4*17)/2;
l2 = -3 + sqrt(36+4*17)/2;

dif = exp(l2) - exp(l1);
tot = exp(l2) + exp(l1);
c1 = dif/(l2-l1);
c0 = exp(l1)-c1*l1;

c = [c0+2*c1,c1;c1,c0-8*c1]

[V,D,W] = eig(A);
for i=1:2
    D(i,i) = exp(D(i,i));
end
d = (W'^-1)*D*(W')

e = expm(A)
%% Problem 8 new
l1 = -1;
l2 = 5;

dif = exp(l2) - exp(l1);
tot = exp(l2) + exp(l1);
c1 = dif/(l2-l1)
c0 = exp(l1)-c1*l1

%% Problem 9 new
c1 = -sin(1)*sin(2)
c0 = cos(2)*cos(1)-2*c1
A = [2 1;-1 2];
cA = c1*A+c0*eye(2)

%% Problem 10 new
A = [2 1 0 0 0;
    0 2 1 0 0;
    0 0 2 0 0;
    0 0 0 2 1;
    0 0 0 0 1];
A = jordan(A)