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    %% Eigenvalues of Nondiagonalizable (Defective) Matrix  

%% 
% Create a 3-by-3 matrix. 
A = [3 1 0; 0 3 1; 0 0 3];  

%% 
% Calculate the eigenvalues and right eigenvectors of |A|. 
[V,D] = eig(A) 

%%
% |A| has repeated eigenvalues and the eigenvectors are not independent.
% This means that |A| is not diagonalizable and is, therefore, defective.

%% 
% Verify that |V| and |D| satisfy the equation, |A*V = V*D|, even though
% |A| is defective.
A*V - V*D 

%%
% Ideally, the eigenvalue decomposition satisfies the relationship. Since
% |eig| performs the decomposition using floating-point computations, then
% |A*V| can, at best, approach |V*D|. In other words, |A*V - V*D| is close
% to, but not exactly, |0|.