Let T be a tournament with n vertices v1,…,vn. The skew-adjacency matrix of T is the n×n zero-diagonal matrix S=sij in which sij=−sji=1 if vi dominates vj. It is well-known that the determinant of S ...is zero or the square of an odd integer. Moreover, the principal minors of S are at most 1 if and only if T is a local order. In this paper, we characterize the class of tournaments for which the principal minors of the skew-adjacency matrix do not exceed 9.
N-matrices are real n×n matrices all of whose principal minors are negative. We provide (i) an O(2n) test to detect whether or not a given matrix is an N-matrix, and (ii) a characterization of ...N-matrices, leading to the recursive construction of every N-matrix.
We consider the problem of reconstructing a symmetric matrix from its principal minors, which has several applications in information theory and statistical modeling. We develop a theory of symmetric ...matrices with equal corresponding principal minors based on a simple equivalent property due to Oeding (2011) 10. We then use this theory to provide a method for choosing a canonical representative from the class of all symmetric matrices with specified principal minors. Finally, we provide an efficient algorithm for computing this canonical representative given its principal minors as input.
In this article, we study the following problem. Given a positive integer k, what is the relationship between two matrices with entries in a field K and having equal corresponding principal minors of ...order at most k? Our main theorem improves a result of Loewy 13 for skew-symmetric matrices all of whose off-diagonal entries are nonzero.
The almost-principal rank characteristic sequence (apr-sequence) of an n×n symmetric matrix is introduced, which is defined to be the string a1a2⋯an−1, where ak is either A, S, or N, according as ...all, some but not all, or none of its almost-principal minors of order k are nonzero. In contrast to the other principal rank characteristic sequences in the literature, the apr-sequence of a matrix does not depend on principal minors. The almost-principal rank of a symmetric matrix B, denoted by ap-rank(B), is defined as the size of a largest nonsingular almost-principal submatrix of B. A complete characterization of the sequences not containing an A that can be realized as the apr-sequence of a symmetric matrix over a field F is provided. A necessary condition for a sequence to be the apr-sequence of a symmetric matrix over a field F is presented. It is shown that if B∈Fn×n is symmetric and non-diagonal, then rank(B)−1≤ap-rank(B)≤rank(B), with both bounds being sharp. Moreover, it is shown that if B is symmetric, non-diagonal and singular, and does not contain a zero row, then rank(B)=ap-rank(B).
In previous work Belton et al. (2016) 2, the structure of the simultaneous kernels of Hadamard powers of any positive semidefinite matrix was described. Key ingredients in the proof included a novel ...stratification of the cone of positive semidefinite matrices and a well-known theorem of Hershkowitz, Neumann, and Schneider, which classifies the Hermitian positive semidefinite matrices whose entries are 0 or 1 in modulus. In this paper, we show that each of these results extends to a larger class of matrices which we term 3-PMP (principal minor positive).
In this paper, we study a particular $ n\times n $ matrix $ A = a_{k_{ij}}^n_{i, j = 1} $ and its Hadamard inverse $ A^{\circ (-1)} $, whose entire elements are exponential form $ a_k = ...e(\frac{k}{n}) = e^{\frac{2\pi ik}{n}}, $ where $ k_{ij} = \min(i, j)+1 $. We study determinants, leading principal minor and inversions of $ A, $ $ A^{\circ (-1)} $. Then the defined values of Euclidean norms, $ l_p $ norms and spectral norms of these matrices are presented, rather than upper and lower bounds, which are different from other articles.
In this article we consider the following equivalence relation on the class of all functions of two variables on a set X: we will say that L,M:X×X→C are rescalings if there are non-vanishing ...functions f,g on X such that M(x,y)=f(x)g(y)L(x,y), for any x,y∈X. We give criteria for being rescalings when X is a topological space, and L and M are separately continuous, or when X is a domain in Cn and L and M are sesqui-holomorphic.
A special case of interest is when L and M are symmetric, and f=g only has values ±1. This relation between M and L in the case when X is finite (and so L and M are square matrices) is known to be characterized by the equality of the principal minors of these matrices. We extend this result to the case when X is infinite. As an application we characterize restrictions of isometries of Hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors.
All matrices we consider have entries in a fixed algebraically closed field K. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal ...generated by size t principal minors of a generic matrix, and restrict our attention to locally closed subsets of its vanishing set, given by matrices of a fixed rank. The main result is a computation of the dimension of the locally closed set of n×n rank n−2 matrices whose size n−2 principal minors vanish; this set has dimension n2−n−4.
The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric matrix
∈ 𝔽
is defined as ℓ
· · · ℓ
, where ℓ
∈ {A, S, N} according to whether all, some but not all, or none of the ...principal minors of order
of
are nonzero. Building upon the second author’s recent classification of the epr-sequences of symmetric matrices over the field 𝔽 = 𝔽
, we initiate a study of the case 𝔽= 𝔽
. Moreover, epr-sequences over finite fields are shown to have connections to Ramsey theory and coding theory.
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