A copositive Q-matrix which is not R₀

Abstract

Jeter and Pye gave an example to show that Pang's conjecture, that L₁∩Q ⊂ R₀, is false while Seetharama Gowda showed that the conjecture is true for symmetric matrices. It is known that L₁-symmetric matrices are copositive matrices. Jeter and Pye as well as Seetharama Gowda raised the following question: Is it true C₀∩Q ⊂ R₀? In this note we present an example of a copositive Q-matrix which is not R₀. The example is based on the following elementary proposition: Let A be a square matrix of order n. Suppose R₁=R₂ where R_i stands for the ith row of A. Further suppose A₁₁ and A₂₂ are Q-matrices where A_ii stands for the principal submatrix omitting the ith row and ith column from A. Then A is a Q-matrix.