The Schur complements for $SD{D}_1$ matrices and their application to linear complementarity problems

In this paper we propose a new scaling method to study the Schur complements of $SD{D}_1$ matrices. Its core is related to the non-negative property of the inverse $M$-matrix, while numerically improving the Quotient formula. Based on the Schur complement and a novel norm splitting manner, we establish an upper bound for the infinity norm of the inverse of $SD{D}_1$ matrices, which depends solely on the original matrix entries. We apply the new bound to derive an error bound for linear complementarity problems of $B_1$-matrices. Additionally, new lower and upper bounds for the determinant of $SD{D}_1$ matrices are presented. Numerical experiments validate the effectiveness and superiority of our results.