A contribution to the theory of P-matrices (Q2639929)

A real \(n\times n\) matrix is said to be a P-matrix if all of its principal minors are positive. In the paper two criteria for a real \(n\times n\) matrix to be a P-matrix are proposed. Let \(R_ i(A)\) and \(C_ i(A)\) denote the sums of all off-diagonal entries of the ith row and the ith column of A, respectively. Let \(Q_ i(A)\) be the sum of all entries of the ith column of A and \(m_ i(A)\) be the maximum of all off- diagonal entries of the ith row of A. Define \(M^+(A)=\sum^{n}_{i=1}m_ i^+(A)=\sum^{n}_{i=1}(m_ i(A)+| m_ i(A)|).\) Furthermore, set \(T_ i(A)=(1+\#K(A))M^+(A)-C_ i(A)-\sum_{k\in K(A)}Q_ k(A)\) where \(K(A)=\{s| \quad M^+(A)\geq Q_ s(A)\}\) and {\#}K(A) denotes the number of elements in the set K(A). Then it is proved that if \(a_{ii}>M^+(A)-C_ i(A),\) \(i=1,2,...,n\) or \(a_{ii}>\min \{n m^+_ i(A)-R_ i(A),T_ i(A)\},\) \(i=1,2,...,n\), then A is a P- matrix.