On generators of bounded ratios of minors for totally positive matrices

全正性、复合矩阵与 Dodgson 压缩的充分条件

Sufficient conditions for total positivity, compounds, and Dodgson condensation

本文研究 TP 与 TP_k 矩阵的第 r 复合矩阵及 Dodgson 压缩矩阵的全正性，给出一般否定结果，并证明 TP Hankel 矩阵的所有压缩矩阵均为 TP。

Interlacing properties of the eigenvalues of some matrix classes

We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any its principal submatrix) for the class of matrices, introduced by Kotelyansky (all principal and all almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices also possess this property. We prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.

Sign non-reversal property for totally non-negative and totally positive matrices, and testing total positivity of their interval hull

A matrix $A$ is totally positive (or non-negative) of order $k$, denoted $TP_k$ (or $TN_k$), if all minors of size $\leq k$ are positive (or non-negative). It is well-known that such matrices are characterized by the variation diminishing property together with the sign non-reversal property. We do away with the former, and show that $A$ is $TP_k$ if and only if every submatrix formed from at most $k$ consecutive rows and columns has the sign non-reversal property. In fact this can be strengthened to only consider test vectors in $\mathbb{R}^k$ with alternating signs. We also show a similar ch…

Bounded Ratios of Products of Principal Minors of Positive Definite Matrices

Considered is the multiplicative semigroup of ratios of products of principal minors bounded over all positive definite matrices. A long history of literature identifies various elements of this semigroup, all of which lie in a sub-semigroup generated by Hadamard-Fischer inequalities. Via cone-theoretic techniques and the patterns of nullity among positive semidefinite matrices, a semigroup containing all bounded ratios is given. This allows the complete determination of the semigroup of bounded ratios for 4-by-4 positive definite matrices, whose 46 generators include ratios not implied by Had…

Algebras with ternary law of composition and their realization by cubic matrices

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions we…

Factorizations of $k$-Nonnegative Matrices

A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k = n-1$ and when $k = n-2$, restricted to unitriangular matrices. For these two cases, we prove that the set of $k$-nonnegative matrices can be partitioned into cells based on their factorizations into generators, generalizing the notion of Bruhat cells from totally nonnegative matrices. Like Bruhat cells, these cells are homeomorphic to open balls and have a to…

全负性：刻画与单向量检验

Total negativity: Characterizations and single-vector tests

本文利用符号不反转、变差消减和线性互补问题给出全负矩阵的单检验向量刻画，并将其用于矩阵区间包络的全负性检验。

振荡矩阵的次阶指数轮廓

Minor-Order Exponent Profiles of Oscillatory Matrices

本文确定振荡矩阵各阶子式首次全正时间的轮廓，提出两角准则，证明尖锐不等式，并给出基于相邻双对角分解的有限刻画与整数幺模实现。