Olga Holtz, Bernd Sturmfels
The principal minors of a symmetric $n{\times}n$-matrix form a vector of length $2^n$. We characterize these vectors in terms of algebraic equations derived from the $ 2{\times}2{\times}2$-hyperdeterminant.
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We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.
On the signs of the principal minors of Hermitian matrices
本文定义了埃尔米特矩阵的带符号增强主秩特征序列，并完整刻画了不作为任何埃尔米特矩阵及实对称矩阵该序列子序列出现的2阶和3阶序列。
On the principal minors of the powers of a matrix
本文证明矩阵各次幂的对角元由原矩阵的主子式唯一确定，并可表示为后者的通用整系数多项式；若所有主子式均为1，则对角元也为1。
Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment
本文研究线性主子式多项式，建立其与多仿射稳定多项式的对应，推广Fisher-Hadamard与Koteljanskii不等式至PSD稳定情形，并猜想谱包含关系。
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…
Principal matrices of a numerical semigroup of embedding dimension n are special types of $n \times n$ matrices over integers of rank $\leq n - 1$. We show that such matrices and even the pseudo principal matrices of size n must have rank $\geq \frac{n}{2}$ regardless of the embedding dimension. We give structure theorems for pseudo principal matrices for which at least one $n - 1 \times n - 1$ principal minor vanish and thereby characterize the semigroups in embedding dimensions $4$ and $5$ in terms of their principal matrices. When the pseudo principal matrix is of rank $n - 1$, we give a su…
A singular M-matrix perturbed by a nonnegative rank one matrix has positive principal minors; is it D-stable?
研究奇异M矩阵在被非负秩一扰动后的正稳定性与D稳定性，特别情况下可以证明正稳定性或D稳定性，但一般情况无法成立，通过反例说明。然而，此类矩阵被证明为P矩阵。
We provide a method for factoring all bounded ratios of the form $$\det A(I_1|I_1')\det A(I_2|I_2')/\det A(J_1|J_1')\det A(J_2|J_2')$$ where $A$ is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera's result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors.
The principal pivot transform (PPT) of a matrix A partitioned relative to an invertible leading principal submatrix is a matrix B such that A [x_1^T x_2^T]^T = [y_1^T y_2^T]^T if and only if B [y_1^T x_2^T]^T = [x_1^T y_2^T]^T, where all vectors are partitioned conformally to A. The purpose of this paper is to survey the properties and manifestations of PPTs relative to arbitrary principal submatrices, make some new observations, present and possibly motivate further applications of PPTs in matrix theory. We pay special attention to PPTs of matrices whose principal minors are positive.
The principal minors of the Toeplitz matrix $\left( x_{i-j+1}\right)_{1\le i,j,\le n}$, where $x_0=1, x_k=0$ if $k\le -1$, directly determine an involution of the polynomial ring $R[x_1, ... ,x_n]$ over any commutative ring $R$.
A common task in physics, information theory, and other fields is the analysis of properties of subsystems of a given system. Given the covariance matrix $M$ of a system of $n$ coupled variables, the covariance matrices of the subsystems are principal submatrices of $M$. The rapid growth with $n$ of the set of principal submatrices makes it impractical to exhaustively study each submatrix for even modestly-sized systems. It is therefore of great interest to derive methods for approximating the distributions of important submatrix properties for a given matrix. Motivated by the importance of di…
The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence restrict to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads t…
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.
Positivity preservers over finite fields II
本文解决了有限域上逐元素保正算子分类中$n=2$、$q\boldsymbol{\text{≡}}1\boldsymbol{\text{(mod}\text{ }4\text{)}}$且$q$非平方数的遗留情形，结合幂等归约与二次特征性质完成了所有有限域及固定维数的完整分类。
A stronger Sylvester's criterion for positive semidefinite matrices
本文提出一个更强的正半定矩阵西尔维斯特判据，将所需验证的行列式数量从指数级 $2^m-1$ 降至 $m(m+1)/2$，并基于新判据给出逐元素判据及其在矩阵填充和非线性半定规划中的应用。
N-matrices are real $n\times n$ matrices all of whose principal minors are negative. We provide (i) an $O(2^n)$ 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.
Is This a New Class of Matrices?
本文通过符号矩阵共轭构造了一类与实方阵相关的新矩阵，证明其与原矩阵相似相合且积和式多项式相同，并研究了映射群结构、对称分解及不同像矩阵的计数公式。
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