Johnson, An Open Problem And A Schur Complement Inequality

This paper consists of two parts. In the first part we discuss an open question concerning Inverse M-matrices given in C. R. Johnson's paper [3]. We introduce and investigate another question, which is essentially equivalent to Johnson's original question, then the question is generalized and a positive answer is given. The second part of this paper is mainly concerned about an interesting matrix inequality presented in [5], which is then generalized in M~' under the entry-wise nonnegative ordering. We introduced the concept of sub-Kronecker product, and establish an inequality which relates the Schur complement of A and B for positive semidefinite matrices A and B. Our results improve the related known results obtained by T. L. Markham and R. L. Smith in 1998 (see [5]).