% eigen.m  demonstrate eigenvalues in MatLab
format compact
e1=1; % desired eigenvalues
e2=2;
e3=3;
e4=4;
P=poly([e1 e2 e3 e4]) % make polynomial from roots
r=roots(P) % check that roots come back
A=zeros(4);
A(1,1)=-P(2); % build matrix A
A(1,2)=-P(3);
A(1,3)=-P(4);
A(1,4)=-P(5);
A(2,1)=1;
A(3,2)=1;
A(4,3)=1
[v e]=eig(A) % compute eigenvectors and eigenvalues
I=eye(4);    % identity matrix
for i=1:4
  z1=det(A-I.*e(i,i)) % should be near zero
end
for i=1:4
  z2=A*v(:,i)-e(i,i)*v(:,i) % note columns, near zero
end
z3=trace(A)-trace(e) % should be near zero

%  annotated output
%
%P =
%     1   -10    35   -50    24
%r =
%    4.0000         these should be eigenvalues
%    3.0000
%    2.0000
%    1.0000
%A =
%    10   -35    50   -24   polynomial coefficients
%     1     0     0     0
%     0     1     0     0
%     0     0     1     0
%v =
%    0.9683    0.9429    0.8677    0.5000  eigenvectors
%    0.2421    0.3143    0.4339    0.5000  are
%    0.0605    0.1048    0.2169    0.5000  columns
%    0.0151    0.0349    0.1085    0.5000
%e =
%    4.0000         0         0         0  eigenvalues
%         0    3.0000         0         0  as diagonal
%         0         0    2.0000         0  of the matrix
%         0         0         0    1.0000
%z1 =
%   1.1280e-13   i=1 first eigenvalue
%z1 =
%   5.0626e-14
%z1 =
%   1.3101e-14
%z1 =
%   2.6645e-15
%z2 =
%   1.0e-14 *    note multiplier  i=1
%       -0.4441
%       -0.1110
%       -0.0333
%       -0.0035
%z2 =
%   1.0e-14 *
%       -0.4441
%       -0.1554
%       -0.0444
%       -0.0042
%z2 =
%   1.0e-14 *
%       -0.7994
%       -0.2776
%       -0.0833
%       -0.0028
%z2 =
%   1.0e-13 *
%       -0.3103
%       -0.0600
%       -0.0122
%       -0.0033
%z3 =
%  -7.1054e-15     trace check