Incremental spectral clustering by efficiently updating the eigensystem

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I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. i want to know if it is possible to have eigenvalues of matrix $A uv^T$. $$ As you see i found the solution, but after implementation of this i can see a very small error in result, it would be appreciated if anyone know why this is happening, because we didn't used any approximation to get the result and it should be exact.I don't need its eigenvectors, but it is required to have the most precise eigenvalues. Found the solution in [1], Please let me know if i had any mistake.

Quantitative measures of change based on feature organization: Eigenvalues and eigenvectors. Proceedings CVPR'96, 1996 IEEE Computer Society Conference on.

Finally, the good performance of this algorithm can be proved by experiment calculation and concrete examples.

For the operation, 1-move and 2-opt were applied, it can also fasten the speed of convergence, and boost the search efficiency.

Based on the analysis of the fault characteristics during the process of active sludge sewage treatment, a PSO clustering algorithm is presented.

In Richardson T and Jaakkola T, editors, Workshop on Artificial Intelligence and Statistics, Society for Artificial Intelligence and Statistics, Key West, FL(2001). Abstract: Various unusual conditions are likely to occur during sewage treatment process, which would lead to some consequences such as the decrease of water quality in the process of sewage treatment and the increase of disposal process, whereby causing a great influence to the practical operation efficiency of sewage treatment factories.

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