disp('Homework problems for section 6.2, Fall 1996, MAT 342')
disp('')
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
clear 
disp('Problem 4:')
disp('')
A=[-1 1; 1 1]/sqrt(2)
p=poly(A)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-10,10],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
[EVects,EVals]=eig(A)
disp('Check: Multiply EVects*EVals*inv(EVects) and compare to the original matrix A')
pause
A
check=EVects*EVals*inv(EVects)
pause

clear 
disp('Problem 8:')
disp('')
A=[1 0 3;2 1 2;3 0 1]
p=poly(A)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-20,20],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-20,20])
[EVects,EVals]=eig(A)
disp('Check: Multiply EVects*EVals*inv(EVects) and compare to the original matrix A')
pause
A
check=EVects*EVals*inv(EVects)
pause

clear 
disp('Problem 9:')
disp('')
A=[8 9 9;3 2 3;-9 -9 -10]
p=poly(A)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-10,10],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([-10,10,-10,10])
disp('The matrix A has a repeated eigenvalue at s=-1.')
disp('We row-reduce the matrix (sI-A) and check its RANK:')
reducedA=rref(-1*eye(3)-A)
disp('Clearly there is a TWO-parameter family of solutions, i.e. ')
disp('there are INDEPENDENT eigenvectors associated to this double')
disp('eigenvalue, and hence the matrix A is similar to a diagonal')
disp('matrix.')
pause

[EVects,EVals]=eig(A)
disp('Check: Multiply EVects*EVals*inv(EVects) and compare to the original matrix A')
pause
A
check=EVects*EVals*inv(EVects)

clear 
disp('Problem 14:')
disp('')
A=[2 1 0;0 2 0;0 0 3]
p=poly(A)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-10,10],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([1,4,-1,1])
pause
disp('The matrix A has a repeated eigenvalue at s=2.')
disp('We row-reduce the matrix (sI-A) and check its RANK:')
reducedA=rref(2*eye(3)-A)
disp('Clearly there is only a ONE-parameter family of solutions')
disp('i.e. there is only one eigenvector associated to this double')
disp('eigenvalue, and hence the matrix A is not similar to a diagonal')
disp('matrix.')
pause
disp('For comparison, we take a look what MATLAB''s eig returns:')
pause
[EVects,EVals]=eig(A)
disp('As expected, MATLAB can''t find INDEPENDENT eigenvectors either!')
pause
A
check=EVects*EVals*inv(EVects)

clear 
disp('Problem 16:')
disp('')
A=[2 0 1;3 3 3;1 0 2]
p=poly(A)
clf
ss=roots(p);
xx=[-10:.1:10];
yy=polyval(p,xx);
plot([-10,10],[0,0],'g-',[0,0],[-10,10],'c-');
hold on
plot(xx,yy,'m',ss,zeros(size(ss)),'yo');
axis([0,5,-2,2])
disp('The matrix A has a repeated eigenvalue at s=3.')
disp('We row-reduce the matrix (sI-A) and check its RANK:')
reducedA=rref(3*eye(3)-A)
disp('Again there is only a ONE-parameter family of solutions')
disp('i.e. there is only one eigenvector associated to this double')
disp('eigenvalue, and hence the matrix A is not similar to a diagonal')
disp('matrix.')
pause
disp('For comparison, we take a look what MATLAB''s eig returns:')
pause
[EVects,EVals]=eig(A)
disp('As expected, MATLAB can''t find INDEPENDENT eigenvectors either!')