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Iterative Methods for Computing Eigenvalues and Eigenvectors

The Waterloo Mathematics Review 9. Iterative Methods for Computing Eigenvalues and Eigenvectors Maysum Panju University of Waterloo Abstract: We examine some numerical Iterative Methods for Computing the Eigenvalues and eigenvec- tors of real matrices. The five Methods examined here range from the simple power iteration method to the more complicated QR iteration method. The derivations, procedure, and advantages of each method are briefly discussed. 1 Introduction Eigenvalues and Eigenvectors play an important part in the applications of linear algebra. The naive method of finding the Eigenvalues of a matrix involves finding the roots of the characteristic polynomial of the matrix. In industrial sized matrices, however, this method is not feasible, and the Eigenvalues must be obtained by other means. Fortunately, there exist several other techniques for finding Eigenvalues and Eigenvectors of a matrix, some of which fall under the realm of Iterative Methods .

Methods for Computing Eigenvalues and Eigenvectors 10 De nition 2.2. The characteristic polynomial of A , denoted P A (x ) for x 2 R , is the degree n polynomial de ned by P A (x ) = det( xI A ): It is straightforward to see that the roots of the characteristic polynomial of a …

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