Matrices and Matroids for Systems Analysis
Matrices and Matroids for Systems Analysis
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DOI：
10.1007/978-3-642-03994-2
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发表时间：
2000-01
期刊：
影响因子：
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通讯作者：
K. Murota
中科院分区：
文献类型：
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作者：
K. Murota
矩阵理论和矩阵理论之间的相互作用是这本书的主题，它提供了一个矩阵理论的方法来线性代数和，反过来，线性代数的方法来矩阵理论。本书还作为理论和混合矩阵和混合多项式矩阵的应用的第一个全面介绍。矩阵是一种抽象的数学结构，它捕获了矩阵的组合性质，而矩阵的组合性质反过来又可以借助矩阵理论成功地表述和分析。在拟阵理论中最重要、数学内容最深刻、应用最有用的一个结果是交点定理，即一对拟阵的对偶定理。同样，多项式矩阵的组合性质也可以用有值拟阵的语言来表述，并且，交定理也可以推广到一对有值拟阵。混合矩阵的概念是在八十年代初形成的，作为一种数学工具，通过矩阵理论组合方法进行系统分析。如果矩阵表示为具有代数无关的非零项的“常数”矩阵和“一般”矩阵的和，则称为混合矩阵。这个概念是由物理观察得出的，在工程系统的描述中，需要区分两种不同的数字，固定常数和系统参数。混合矩阵的数学分析可以通过应用于与“常数”和“一般”矩阵相关的阵对的交点定理来简化。根据求值矩阵的交点定理，这种方法可以进一步推广到混合多项式矩阵。
Interplay between matrix theory and matroid theory is the main theme of this book, which offers a matroid-theoretic approach to linear algebra and, reciprocally, a linear-algebraic approach to matroid theory. The book serves also as the first comprehensive presentation of the theory and application of mixed matrices and mixed polynomial matrices. A matroid is an abstract mathematical structure that captures combinatorial properties of matrices, and combinatorial properties of matrices, in turn, can be stated and analyzed successfully with the aid of matroid theory. The most important result in matroid theory, deepest in mathematical content and most useful in application, is the intersection theorem, a duality theorem for a pair of matroids. Similarly, combinatorial properties of polynomial matrices can be formulated in the language of valuated matroids, and moreover, the intersection theorem can be generalized for a pair of valuated matroids.The concept of a mixed matrix was formulated in the early eighties as a mathematical tool for systems analysis by means of matroid-theoretic combinatorial methods. A matrix is called a mixed matrix if it is expressed as the sum of a “constant” matrix and a “generic” matrix having algebraically independent nonzero entries. This concept is motivated by the physical observation that two different kinds of numbers, fixed constants and system parameters, are to be distinguished in the description of engineering systems. Mathematical analysis of a mixed matrix can be streamlined by the intersection theorem applied to the pair of matroids associated with the “constant” and “generic” matrices. This approach can be extended further to a mixed polynomial matrix on the basis of the intersection theorem for valuated matroids.