论文标题
巴雷特·约翰逊（Barrett-Johnson）完全非负矩阵的不平等现象
Barrett-Johnson inequalities for totally nonnegative matrices
论文作者
论文摘要
给定一个矩阵$ a $，令$ a_ {i，j} $表示$ a $的combatrix由$ i $和列$ j $确定。 Fischer的不平等表明，对于每个$ n \ times n $ hermitian阳性半矩阵$ a $ a $，每个子集$ i $的$ \ {1，\ dotsc，n \} $及其补充$ i^c $ \ det（a_ {i，i}）\ det（a_ {i^c，i^c}）$。 Barrett和Johnson（线性多线性代数34，1993）将其扩展到规定不平等的主要未成年人产物的总和，其订单的订单是通过非插入整数序列$（λ_1，\ dotsc，λ_r）$，$（μ_1，$ __1，\ dotsc，\ dotsc，\ dotsc，\ dotsc，μ__s）$ n $ n。具体来说，如果$λ_1+\ cdots+λ_i\leqμ_1+\ cdots+μ_i$用于所有$ i $，则是$λ_1！\cdotsλ_r！ \ sum _ {（i_1，\ dotsc，i_r）} \ det（a_ {a_ {i_1，i_1}）\ cdots \ det（a__ {i_r，i_r，i_r}）〜\ geq〜 μ_1！\cdotsμ_s！ \ sum _ {（J_1，\ dotsc，j_s）} \ det（a_ {j_1，j_1}）\ cdots \ det（a_ {a_ {j_s，j_s}），$$，$ \ {1，\ dotsc，n \ dotsc，n \} $满意=λ_k$，$ | j_k | =μ_k$。我们表明，这些不平等也适用于完全非负矩阵。
Given a matrix $A$, let $A_{I,J}$ denote the submatrix of $A$ determined by rows $I$ and columns $J$. Fischer's Inequalities state that for each $n \times n$ Hermitian positive semidefinite matrix $A$, and each subset $I$ of $\{1,\dotsc,n\}$ and its complement $I^c$, we have $\det(A) \leq \det(A_{I,I})\det(A_{I^c,I^c})$. Barrett and Johnson (Linear Multilinear Algebra 34, 1993) extended these to state inequalities for sums of products of principal minors whose orders are given by nonincreasing integer sequences $(λ_1,\dotsc,λ_r)$, $(μ_1,\dotsc,μ_s)$ summing to $n$. Specifically, if $λ_1+\cdots+λ_i\leq μ_1+\cdots+μ_i$ for all $i$, then $$ λ_1!\cdotsλ_r! \sum_{(I_1,\dotsc,I_r)} \det(A_{I_1,I_1}) \cdots \det(A_{I_r,I_r}) ~\geq~ μ_1!\cdotsμ_s! \sum_{(J_1,\dotsc,J_s)} \det(A_{J_1,J_1}) \cdots \det(A_{J_s,J_s}), $$ where sums are over sequences of disjoint subsets of $\{1,\dotsc,n\}$ satisfying $|I_k| = λ_k$, $|J_k| = μ_k$. We show that these inequalities hold for totally nonnegative matrices as well.
