The research of M-matrix involving Fan product is an important topic of M-matrix theory research. Three inequalities on the minimum eigenvalue of Fan product are exhibited through the Perron eigenvectors of the M-matrices. The inequalities can be regarded as useful complements to existing research results.

In the areas of statistics, matrix analysis, and several applications of applied mathematics, the Perron complement is fundamental. For a matrix with nonnegative elements, some inequalities and equalities involving generalized Perron complement and Perron complement of the matrices are derived.

For an m × n sign pattern P, we define a signed bipartite graph B ( U , V ) with one set of vertices U = { 1 , 2 , … , m } based on rows of P and the other set of vertices V = { 1 ′ , 2 ′ , … , n ′ } based on columns of P. The zero forcing number is an important graph parameter that has been used to study the minimum rank problem of a matrix. In this paper, we introduce a new variant of zero forcing set−bipartite zero forcing set and provide an algorithm for computing the bipartite zero forcing number. The bipartite zero forcing number provides an upper bound for the maximum nullity of a square full sign pattern P. One advantage of the bipartite zero forcing is that it can be applied to study the minimum rank problem for a non-square full sign pattern.

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For an irreducible and nonpositive matrix, we present the concepts of the Perron complement matrix and the generalized Perron complement matrix. An inequality that relates the generalized Perron complement matrices of inverse N 0-matrices is derived. In addition, we obtain the quotient formula of inverse N 0-matrices based on the quotient formula for the Schur complements.