Semipositive matrix, minimally semipositive matrix, principal pivot transform, Moore-Penrose inverse, left inverse, interval of matrices
Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this article, this notion is revisited and new results are presented. It is shown that the set of all $m \times n$ minimally semipositive matrices contains a basis for the linear space of all $m \times n$ matrices. Apart from considerations involving principal pivot transforms and the Schur complement, results on semipositivity and/or minimal semipositivity for the following classes of matrices are presented: intervals of rectangular matrices, skew-symmetric and almost skew-symmetric matrices, copositive matrices, $N$-matrices, almost $N$-matrices and almost $P$-matrices.
Choudhury, Projesh Nath; Kannan, Rajesh M.; and Sivakumar, K. C..
"New Contributions to Semipositive and Minimally Semipositive Matrices",
Electronic Journal of Linear Algebra,
Volume 34, pp. 35-53.