Study on Eigen Values And Eigen Vectors of Matrices: An Iterative Approach

Abstract

There are some smart methods, available in the literature, which are able to find out all Eigen values. But those methods could not find corresponding Eigen vectors simultaneously. Power method and Inverse Power method are able to find out both Eigen-pairs simultaneously. Power method frequently used for finding only largest Eigen value and corresponding Eigen vector. On the other hand Inverse Power method is applied to find out only smallest Eigen value (or desire Eigen value) and corresponding Eigen vector. But Inverse Power method is computationally costly and some time it is unstable for the presence of inverse of the matrix. It is theoretically observed that if all Eigen values are either positive or negative, then without implement of Inverse Power method the modified (using shifting property) Power method is also able to find out smallest Eigen-pair. Here we have proposed Modified Hybrid Iterative Algorithm based on both Power method and Inverse Power method respectively to find out both largest and smallest Eigen-pairs simultaneously. Moreover several lemma regarding the proposed algorithm have been proposed. The proof of each lemma has also been given along with some suitable illustration. Several experiments have been performed to investigate the robustness and effectiveness of the algorithm as well as the lemma. The proposed algorithm is able to find out both (largest and smallest) Eigen-pairs successfully as well as efficiently.