Some properties of fully semimonotone Q~0-matrices

Abstract

Stone [Ph.D. thesis, Dept. of Operations Research, Stanford University, Stanford, CA, 1981] proved that within the class of $Q_0 $-matrices, the $U$-matrices are $P_0 $-matrices and conjectured that the same must be true for fully semimonotone $(E_0^f )$ matrices. In this paper we show that this conjecture is true for matrices of order up to $4 \times 4$ and partially resolve it for higher order matrices. This is done by establishing the result that if $A$ is in $E_0^f \cap Q_0 $ and if every proper principal minor of $A$ is nonnegative, then $A$ is a $P_0 $-matrix. Using this key result we settle the conjecture for a number of special cases of matrices of general order. These special cases include $E_0^f $-matrices which are either symmetric or nonnegative or copositive-plus or $Z$-matrices or $E$-matrices. Also the conjecture is established for $5 \times 5$ matrices with all diagonal entries positive. While trying to settle the conjecture, we obtained a number of results on $Q_0 $-matrices. The main among these are characterizations of nonnegative $Q_0 $-matrices and symmetric semimonotone $Q_0 $-matrices; results providing sufficient conditions under which, principal submatrices of order $(n - 1)$ of a $n \times n$$Q_0 $-matrix are also in $Q_0 $.