Total nonpositivity of nonsingular matrices (Q967509)

The authors present new characterizations of nonsingular totally nonpositive matrices. These characterizations are described in terms of minors with consecutive initial rows or consecutive initial columns. The main result of this work establishes that if \(A=(a_{ij})\) is a nonsingular real matrix, of size \(n \times n\), \(n \geq 2\), the following statements are equivalent: (1) \(A\) is totally nonpositive. (2) For any \(k \in \{1,2,\dots,n-1 \}\), \[ \begin{aligned} &a_{11} \leq 0, \quad a_{nn} \leq 0, \quad a_{n1} < 0, \quad a_{1n} < 0, \\ &\det{A[\alpha | \{k+1,\dots,n \}]} \leq 0, \quad \forall \alpha \in Q_{n-k,n}, \\ &\det{A[\{k+1,\dots,n \} | \beta]} \leq 0, \quad \forall \beta \in Q_{n-k,n}, \\ &\det{A[\{k,\dots,n \}]} < 0. \end{aligned} \] (3) For any \(k \in \{1,2,\dots,n-1 \}\), \[ \begin{aligned} &a_{11} \leq 0, \quad a_{nn} \leq 0, \quad a_{n1} < 0, \quad a_{1n} < 0, \\ &\det{A[\alpha | \{1,\dots,k \}]} \leq 0, \qquad \forall \alpha \in Q_{k,n}, \\ &\det{A[\{1,\dots,k \} | \beta]} \leq 0, \qquad \forall \beta \in Q_{k,n}, \\ &\det{A[\{1,\dots,k+1 \}]} < 0, \end{aligned} \] where \(Q_{k,n}\) denotes the set of strictly increasing sequences of \(k\) natural numbers less than or equal to \(n\).