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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|.