Let V be a nonempty finite set and A = (a ij ) i,j∈V a matrix with entries in a field K. For a subset X of V , we denote by A[X] the submatrix of A having row and column indices in X. In this article, 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 K 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 is an improvement of a result of R. Lowey [13] for skew-symmetric matrices whose all off-diagonal entries are nonzero.

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The idiosyncratic polynomial of a graph G with adjacency matrix A is the characteristic polynomial of the matrix A + y(J − A − I), where I is the identity matrix and J is the all-ones matrix. It follows from a theorem of Hagos (2000) combined with an earlier result of Johnson and Newman (1980) that the idiosyncratic polynomial of a graph is reconstructible from the multiset of the idiosyncratic polynomial of its vertex-deleted subgraphs. For a digraph G with adjacency matrix A, we define its idiosyncratic polynomial as the characteristic polynomial of the matrix A + y(J − A − I) + zA T . By forbidding two fixed digraphs on three vertices as induced subdigraphs, we prove that the idiosyncratic polynomial of a digraph is reconstructible from the multiset of the idiosyncratic polynomial of its induced subdigraphs on three vertices. As an immediate consequence, the idiosyncratic polynomial of a tournament is reconstructible from the collection of its 3-cycles. Another consequence is that all the transitive orientations of a comparability graph have the same idiosyncratic polynomial.

Abstract. It is well known that two n × n symmetric matrices have equal corresponding principal minors of all orders if and only if they are diagonally similar. This result cannot be extended to arbitrary matrices. The aim of this work is to give a new transformation that preserves principal minors of skew-symmetric matrices.