disp('The following are reference solutions and selected comments. ')
disp('Students should continue to work the problems in small steps.')
clear
disp('Note that MATLAB automatically normalizes the eigenvectors by')
disp('making them unit vectors -- since the text book problems are')
disp('"COOKED UP" to give nice numbers, try to find some eigen vectors')
disp('that have integer components');
pause 
A9=[2 0 0 0;0 -1 0 5;0 4 6 4;0 5 0 -1]
p=poly(A9)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-100,100],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-100,100])
[EVectors,EVals]=eig(A9)
pause
A22=[4 1 0;1 4 0;0 -3 0]
p=poly(A22)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-100,100],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
[EVectors,EVals]=eig(A22)
pause
disp('The following matrix has a repeated eigenvalue, which has two')
disp('"LINEARLY INDEPENDENT" associated eigenvalues -- there are many choices!')
pause
A26=[0 0 2 0;1 0 1 0;0 1 -2 0;0 0 0 1]
p=poly(A26)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-100,100],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
[EVectors,EVals]=eig(A26)
pause
disp('The following three problems adress matrices that have ones on')
disp('the sinistral diagonal bottom left to top rite. Again, there are')
disp('repeated eigenvalues, leading to several eigenvectors for some evalues.')
pause
A51=flipud(eye(2))
p=poly(A51)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-100,100],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
[EVectors,EVals]=eig(A51)
pause
A52=flipud(eye(3))
p=poly(A52)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-100,100],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
[EVectors,EVals]=eig(A52)
pause
A53=flipud(eye(4))
p=poly(A53)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-100,100],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
[EVectors,EVals]=eig(A53)
pause
A58=[1 0 0 0 0 -3;2 2 0 0 0 -1;3 0 1 0 0 4;...
     4 0 0 -1 0 3;-1 0 0 0 1 2;-3 0 0 0 0 1]
p=poly(A58)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-100,100],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
[EVectors,EVals]=eig(A58)
pause
hw62