function [] = prob_10_6
% PROB_10_6 - 
% 
% Written by:
% -- 
% John L. Weatherwax                2005-08-28
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

% Computing the right and left eigenvalues exactly: 
%
A = gallery(5);
[x,D] = eig( sym(A) );
[y,D] = eig( sym(A.') ); 

% Since the eigenvalue condition number is given by ||x|| ||y|| / y^H x
% first compute y^H x
y' * x
% Since this is zero the eigenvalue condition estimate should be infinite.

% Due to rounding errors the computation of the right and left eigen values 
% will not be exact:
[X,D] = eig( A ); 
[Y,D] = eig( A.' ); 

% One method of showing the numerical eigenvalue condition number is ~10^10
n1 = diag( X' * X ); n2 = diag( Y' * Y ); 
d1 = diag( Y' * X ) 

( n1 .* n2 ) ./ d1

% Another method: 
condeig( A )