Skew-symmetric matrices and their principal minors

2014/03/01 by Abderrahim Boussaïri, Boussairi, A., Brahim Chergui +1
Computer Science · Engineering · #Combinatorics (math.CO) #FOS: Mathematics #Matrix Theory and Algorithms #Polynomial and algebraic computation #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1403.0095

openalex publication_date 2014/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V be a nonempty finite set and A=(a_ij)_{i,j∈ V} be a matrix with entries in a field \mathbbK. For a subset X of V, we denote by A[X] the submatrix of A having row and column indices in X. We study the following problem. Given a positive integer k, what is the relationship between two matrices A=(a_ij)_{i,j∈ V}, B=(b_ij)_{i,j∈ V} with entries in \mathbbK and such that det(A[ X])=det(B[ X]) for any subset X of V of size at most k ? The Theorem that we get in this Note is an improvement of a result of R. Loewy [5] for skew-symmetric matrices whose all off-diagonal entries are nonzero.

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