Laplace's Theorem in semirings
首发时间:2011-11-04
Abstract:Laplace's Theorem is an important theorem using forcalculating the determinant of matrix in classical linear algebras.This paper tries to generalize it and studies it over semirings, itfirst introduces the notions of a k-order minor and its algebraiccofactor, respectively, then provides an extension of Laplace'sTheorem over commutative semirings with the help of biderminant,i.e., the bideterminant det(D) of matrix D equals to the sum ofall k-order minors det(Mi) multiply their correspondingalgebraic cofactors det(Ai), where 1≤n≤n-1. In the end,we obtain some properties of the bideterminant for matrices oversemirings and show that the bideterminant of an upper triangularblock matrix equals to the product of all the bideterminants ofdiagonal elements.
keywords: Semiring Bideterminant Laplace's Theorem
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半环上的Laplace定理
摘要:Laplace定理是经典线性代数中用于计算矩阵行列式的重要定理,本文试图推广该定理,研究交换半环上Laplace定理。首先给出矩阵的k-级子式与其代数余子式的定义,讨论了交换半环上矩阵的$k$-级子式与其代数余子式的性质,然后借助矩阵的双行列式给出了交换半环上矩阵的Laplace定理:即,交换半环上n阶矩阵D的双行列式等于矩阵D的所有k-级子式与其对应的代数余子式之积之和,其中1≤n≤n-1。在此基础上给出了交换半环上矩阵双行列式展式的性质,最后证明了分块上三角型矩阵的双行列式等于对角线上所有矩阵双行列式之积。
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