© 2016 Charles R. Johnson and Robert B. Reams, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Spec. Matrices 2016; 4:67ś72 Research Article Open Access Charles R. Johnson and Robert B. Reams* Sufcient conditions to be exceptional DOI 10.1515/spma-2016-0007 Received April 22, 2015; accepted November 12, 2015 Abstract: A copositive matrix A is said to be exceptional if it is not the sum of a positive semidefnite matrix and a nonnegative matrix. We show that with certain assumptions on A −1 , especially on the diagonal entries, we can guarantee that a copositive matrix A is exceptional. We also show that the only 5-by-5 exceptional matrix with a hollow nonnegative inverse is the Horn matrix (up to positive diagonal congruence and permutation similarity). Keywords: copositive matrix; positive semidefnite; nonnegative matrix; exceptional copositive matrix; irre- ducible matrix MSC: 15A18, 15A48, 15A57, 15A63 1 Introduction All of the matrices considered will be symmetric matrices with real entries. We will say a matrix is a nonneg- ative matrix if all of its entries are nonnegative, and likewise for a vector. A symmetric matrix A ∈ R n×n is positive semidefnite (positive defnite) if x T Ax ≥0 for all x ∈ R n (x T Ax > 0 for all x ∈ R n , x ≠ 0). A symmetric matrix A ∈ R n×n is called copositive (strictly copositive) if x T Ax ≥0 for all x ∈ R n , x ≥0 (x T Ax > 0 for all x ∈ R n , x ≥0, x ≠ 0). We will let e i ∈ R n denote the vector with ith component one and all other components zero. A permutation matrix is an n-by-n matrix whose columns are e 1 , ..., e n in some order. For n ≥2, an n-by-n matrix is said to irreducible [9] if under similarity by a permutation matrix, it cannot be written in the form ( A 11 0 A 21 A 22 ) , with A 11 and A 22 square matrices of order less than n. We call an n-by-n matrix hollow if all of its diagonal entries are zero. 2 When the inverse is nonnegative and hollow The results in this paper grew out of a question that arose from studying symmetric, nonnegative, hollow, invertible matrices in [4]. Theorem 1, despite its short proof and the fact that we will extend it in Section 3, is the core theorem of this paper. Theorem 1. Suppose A ∈ R n×n is symmetric, invertible, and that A −1 is nonnegative and hollow. If A is of the form A = P + N, with P positive semidefnite and N nonnegative, then P is zero. *Corresponding Author: Robert B. Reams: Department of Mathematics, SUNY Plattsburgh, 101 Broad Street, Plattsburgh, NY 12901, E-mail: robert.reams@plattsburgh.edu Charles R. Johnson: Department of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187