Interval hulls of N-matrices and almost P-matrices
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We establish a characterization of almost P-matrices via a sign non-reversal property. In this we are inspired by the analogous results for N-matrices. Next, the interval hull of two m × n matrices A=(a_ij) and B = (b_ij), denoted by 𝕀(A,B), is the collection of all matrices C ∈ℝ^m × n such that each c_ij is a convex combination of a_ij and b_ij. Using the sign non-reversal property, we identify a finite subset of 𝕀(A,B) that determines if all matrices in 𝕀(A,B) are N-matrices/almost P-matrices. This provides a test for an entire class of matrices simultaneously to be N-matrices/almost P-matrices. We also establish analogous results for semipositive and minimally semipositive matrices. These characterizations may be considered similar in spirit to that of P-matrices by Bialas-Garloff [Linear Algebra Appl. 1984] and Rohn-Rex [SIMAX 1996], and of positive definite matrices by Rohn [SIMAX 1994].
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