Electronic Transactions on Numerical Analysis. Volume 18, pp. 73-80, 2004. Copyright  2004, Kent State University. ISSN 1068-9613. ETNA Kent State University etna@mcs.kent.edu A NEW GER ˇ SGORIN-TYPE EIGENVALUE INCLUSION SET ∗ LJILJANA CVETKOVIC † , VLADIMIR KOSTIC † , AND RICHARD S. VARGA ‡ Abstract. We give a generalization of a less well-known result of Dashnic and Zusmanovich [2] from 1970, and show how this generalization compares with related results in this area. Key words. Gerˇ sgorin theorem, Brauer Cassini ovals, nonsingularity results. AMS subject classifications. 15A18, 65F15. 1. Introduction. Our interest here is in nonsingularity results for matrices and their equivalent eigenvalue inclusion sets in the complex plane. As examples of this, we have the famous result of Gerˇ sgorin [3]: THEOREM 1. For any A =[a i,j ] ∈ C n×n and for any eigenvalue λ of A, there is a positive integer k in N := {1, 2, ··· ,n} such that |λ − a k,k |≤ r k (A) :=  j∈N\{k} |a k,j |. (1.1) Consequently, if σ(A) denotes the collection of all eigenvalues of A, then σ(A) ⊆ Γ(A) := n  i=1 Γ i (A), where Γ i (A) := {z ∈ C : |z − a i,i |≤ r i (A)}. (1.2) Here, Γ i (A) is the i-th Gerˇ sgorin disk, and Γ(A) is the Gerˇ sgorin set for the matrix A. The equivalent nonsingularity result for this is THEOREM 2. For any A =[a i,j ] ∈ C n×n which is strictly diagonally dominant, i.e., |a i,i | >r i (A) (all i ∈ N ), (1.3) it follows that A is nonsingular. Similarly, there is the following nonsingularity result of Ostrowski [5]: THEOREM 3. For any A =[a i,j ] ∈ C n×n ,n ≥ 2, with |a i,i |·|a j,j | >r i (A) · r j (A) (all i = j in N ), (1.4) it follows that A is nonsingular. Its equivalent eigenvalue inclusion set is the following result of Brauer [ 1]: * Received April 8, 2004. Accepted for publication April 30, 2004. Recommended by Lothar Reichel. † Department of Mathematics and Informatics, Faculty of Science, Novi Sad, Serbia and Montenegro. E-mail: {lila, vkostic}@im.ns.ac.yu. The research of the first author was supported in part by the Republic of Serbia, Ministry of Science, Technologies and Development under Grant No. 1771. ‡ Department of Mathematics, Kent State University, Kent, Ohio, U.S.A. E-mail: varga@math.kent.edu. 73