disp('%')
disp('% Homework check for section 6.1: Eigenvalues')
disp('%')
disp('% This sections uses many technical terms from chapters 3 thru 5.')
disp('% The students are not responsible for this material at this time.')
disp('%')
disp('% Definition: A scalar s is an eigenvalue of the matrix A correspon-')
disp('% ding to the nonzero eigenvector x if Ax=sx.')
disp('% ')
disp('% Algorithm: Calculate the characteristic polynomial det(sI-A)')
disp('% and its roots, which are the eigenvalues. For each eigenvalue s')
disp('% solve the systems of equations Ax=sx for the eigenvectora(s) x.')
disp('%')
disp('% Other noteworthy theorems of thsi section:')
disp('% The product of the eigenvalues (with multiplicities) equals det(A).')
disp('% If U is an invertibe matrix with inverse Uinv, then A and Uinv*A*U')
disp('% have the same eigenvalues.')
pause
disp('%')
disp('The characteristic polynomial is a quadratic')

disp('% Problem 6:')
A=[1 2 4;0 3 5;0 0 2]
disp('Use the command "poly" to calculate the characteristic')
disp('Its coefficients are')
p=poly(A)
pause
disp('Use the command "roots" to calculate its roots, that is')
disp('the eigenvalues of the matrix A')
ss=roots(p)
pause
disp('Since it is so easy, always take a look at the graph of the')
disp('characteristic polynomial -- activitate the figure window!')
xx=[-5:.1:5];
yy=polyval(p,xx);
plot([-5,5],[0,0],'g-',[0,0],[-5,5],'c-');
hold on
plot(xx,yy,'m',ss,[0,0,0],'yo');
axis([-5,5,-5,5])
pause
disp('To solve for the corresponding eigenvectors, rowreduce the')
disp('matrices (sI-A), for each eigenvalue s found above.')
s1=ss(1)
A1=s1*eye(3)-A
R1=rref(A1)
pause
disp('One nontrivial solution for this system is:')
pause
v1=[1;1;0]
disp('We check this by mutliplying our solution by the reduced matrix R1')
pause
check=R1*v1
disp('This is zero within working precision, i.e. the solution is correct')
pause
disp('%')
disp('As an alternative check, calculate A*v1 and compare to s1*v1')
pause
v1
Av1=A*v1
s1v1=s1*v1
pause
disp('We repeat the above steps for the second eigenvalue:')
pause
s2=ss(2)
A2=s2*eye(3)-A
R2=rref(A2)
pause
disp('One nontrivial solution for this system is:')
pause
v2=[6;5;-1]
disp('We check this by mutliplying our solution by the reduced matrix R2')
pause
check=R2*v2
disp('This is zero within working precision, i.e. the solution is correct')
pause
disp('%')
disp('As an alternative check, calculate A*v2 and compare to s2*v2')
pause
v2
Av2=A*v2
s2v2=s2*v2
pause
disp('We repeat the above steps for the third eigenvalue:')
pause
s3=ss(3)
A3=s3*eye(3)-A
R3=rref(A3)
pause
disp('One nontrivial solution for this system is:')
pause
v3=[1;0;0]
disp('We check this by mutliplying our solution by the reduced matrix R3')
pause
check=R3*v3
disp('This is zero within working precision, i.e. the solution is correct')
pause
disp('%')
disp('As an alternative check, calculate A*v3 and compare to s3*v3')
pause
v3
Av3=A*v3
s3v3=s2*v3
pause
disp('Finally we check our work against the built-in "eig" command:')
[EVectors,EVals]=eig(A)
disp('and compare this to our solution: D=diag([s1,s2,s3]) and V=transpose of [v1,v2,v3]')
pause
V=[v1,v2,v3]
D=diag([s1,s2,s3])
v2unit=v2/sqrt(v2'*v2)
disp('We see that up to the ordering and normalization, the solutions agree')
disp('where v2unit=v2/sqrt(v2''*v2) is the unit vector in the direction of v2.')
pause
disp('%')
disp('%')
disp('%')
hw61a