New Approach to Identify Common Eigenvalues of real matrices using Gerschgorin Theorem and Bisection method

In this paper, a new approach is presented to determine common eigenvalues of two matrices. It is based on Gerschgorin theorem and Bisection method. The proposed approach is simple and can be useful in image processing and noise estimation. KeywordsCommon Eigenvalues, Gerschgorin theorem, Bisection method, real matrices. INTRODUCTION Eigenvalues play vary important role in engineering applications. A vast amount of literature is also available for computing eigenvalues of a given matrix. Moreover, various numerical techniques such as bisection method, Newton Raphson method, Regula falsi method etc., are available for computing eigenvalues [3]. These methods are applied in various engineering applications. In practice, for some applications, common eigenvalues of the matrices are required. These eigenvalues can be calculated using above methods. In [5], an algorithm is presented to identify common eigenvalues of a two matrices without computing actual eigenvalues of the matrices. But, this method requires Hessenberg transformation of matrices. While going various literature survery, it is observed that except algorithm as proposed [5], no algorithm is available which can be used to identify common eigenvalues of matrices. In practice identification of common eigenvalues are required in various image processing, control systems, and noise estimation applications. Therefore, in this paper attempt has been made to identify common eigenvalues using Gerschgorin theorem and Bisection method. The proposed approach is improvement over the bisection method for computing common eigenvalues. In this paper, Gerschgorin circles have been drawn for two matrices. Then by selecting intersection area of two matrices, bound under which all real common eigenvalues lying are determined. This improved bound is considered as initial approximation for computing eigenvalues of two system matrices using Bisection method. These are compared with the Bisection method with approximate (trial) approximation. II. GERSCHGORIN THEOREM [1] For a given matrix A of order ( n n × ), let k P be the sum of the moduli of the elements along the th k row excluding the diagonal elements kk a . Then every eigenvalues of A lies inside the boundary of atleast one of the circles kk k a P λ − = (1) Suppose above eq.(1) is for row-wise matrix, then similarly, using Gerschgorin theorem, we can write equation for column wise matrix. The intersection region gives the actual eigenvalues of the matrix A. III. BISECTION METHOD [ 3 ] This is one of the simplest iterative methods and it is based on the property of intervals. To find a root using this method, let the function ( ) f x be continuous between a and b . Suppose ( ) f a is positive and ( ), , 0 f b b a t t = + > is negative. Then there is a root of ( ) 0 f x = lying between a and b . III. PROPOSED APPROACH FOR DETERMINING COMMON EIGENVALUES OF THE MATRICES Suppose there are two matrices A and . B We need to determine common eigenvalues of above matrices. The various steps involved in determining common eigenvalues of the matrices as follows. Step 1: Drawing Gerschgorin circles of matrices A and . B Step 2: Determining intersection region of two matrices. Step 3: Based on intersection region, determining bounds on the real axis in the s-plane. 203 http://sites.google.com/site/ijcsis/ ISSN 1947-5500 (IJCSIS) International Journal of Computer Science and Information Security, Vol. 7, No. 2, 2010 Step 4: Using these bounds, under which all common eigenvalues are lying, we calculate common eigenvalues using Bisection method approach as discussed above. III. NUMERICAL EXAMPLE Consider two matrices as