IJCSNS International Journal of Computer Science and Network Security, VOL.10 No.1, January 2010 66 Manuscript received January 5, 2010 Manuscript revised January 20, 2010 Eigenvalues of Structural Matrices Via Gerschgorin Theorem T. D. Roopamala† and S. K. Katti† †Jayachamarajendra College of Engg., University of Mysore, Mysore, India Summary In this paper, we have presented a simple approach for determining eigenvalues for some class of structural matrices. It is based on Gerschgorin theorem. The main advantage of the proposed method is that there is no need to use time-consuming iterative numerical techniques for determining eigenvalues. The proposed approach is expected to be applicable in various computer science and control system applications. Key words: Eigenvalues, Gerschgorin theorem, structural matrices, trace of the matrix 1. Introduction The concept of stability plays very important role in the analysis of systems. A system can be modeled in state space form [1]. In this state space form, stability can be determined by computing the eigenvalues of the system matrix A. There exist various methods in the literature for the computation of the eigenvalues [2, 3]. Moreover, in engineering applications, some structural matrices have been used and thus their eigenvalues computations are also important. In mathematical literature, we found that that there exists Gerschgorin theorem [4-6], which gives bounds under which, all eigenvalues lie. Now a days, eigenvalues can be calculated easily using Matlab. But, we found that Gerschgorin theorem can be useful for computation of some eignvalues without involving iterative numerical technique and softwares. In this paper, we have presented numerical efficient method for computation of eigenvalues using Gerschgorin theorem for some class of structural matrices. The proposed idea can be useful in various engineering applications. 2. Gerschgorin Theorem 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) 3. Determination of Eigenvalues of Structural Matrices Consider a structural system matrix A as a b − b − ... b − b − a b − ... b − [ ] A = M M M M M (2) M M M M M b − b − b − ... a where, 0, 0, ( 1) a b a n b > > = − (3) Applying above Gerschgorin theorem to above matrix A, we get, 1 n ii ij j i i j a a r λ = ≠ − ≤ = ∑ (4) In above matrix [A] , replace , ( 1) ii j a ar n b = = − (5) So, from eq. (4) and eq.(5), we get, a a λ − ≤ (6) By removing modulus of above equation, we get, ( ) a a λ ± − ≤ (7) So, ( ) a a λ − − ≤ (8) or ( ) a a λ − ≤ (9)