When a matrix is perturbed with a small value of 1e-16, the eigenvectors of the perturbed matrix changes significantly. I would expect that such a small perturbation added to a matrix will not change the solution.
The following code returns different eigenvectors for C and C1
clear all;C=[1. 0. 0.; 0. 1. 0. ; 0. 0. 1.];[Vc,ec]=eig(C)C1=C;C1(1,2)=C1(1,2)+1.e-16;[Vc1,ec1]=eig(C1);
The eigenvectors are as follows:
Vc: [1 0 0 0 1 0 0 0 1]; Vc1: [ 1.0000 -0.4106 0 0 0.9118 0 0 0 1.0000];
The eigen vector matrix Vc1 is ill-conditioned and effectively not invertible.
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