Matrix Analysis and Its Applications to Signal Processing

矩阵几乎出现在数学的各个领域，矩阵的分析往往是问题的关键。然而，矩阵理论目前被归为一个经典的研究对象，而对其更深层次的研究却被忽视。本研究的目的是从泛函分析的观点出发，结合图形组合学，研究矩阵分析及其在信号处理中的应用。(1)T. Ando将Hadamard积作为矩阵空间上的线性算子进行了研究，并给出了其范数关于数值半径的一个刻画。他成功地完成参数化的所有极端点的一些凸集矩阵。根据控制的思想，他建立了与Golden-Thompson不等式互补的不等式。(2)从矩阵分析的观点出发，H. Schneider得到了一个赋权有向图平衡的条件。他还建立了高度和指数之间的关系O型图。(3)用组合数学方法R. Brualdi成功地列举了自对偶码的数量。他还获得了一个基本结果，即所有矩阵与一个共同的图是可逆的。(4)N.永井建立了一种通过连续提取2线线路来构造数字滤波器的方法。(5)T. Nakazi建立了一个极值问题之间的关系，在哈代空间和一个属性的汉克尔算子，一个推广的汉克尔矩阵。(6)Y.中村利用优化将矩阵的极值问题推广到函数空间中的度量问题，并分析了其解的结构。他还建立了一个特殊矩阵乘积的范数公式。

Matrices appear in almost all fields of mathematics, and their analysis is often a key point of the problems. Matrix theory, however, is now classified into a classical research object and its deeper research is ignored. This research aims at matrix analysis from the standpoint of functional analysis and in relation to graph-combinatorics, and its application to signal processing.(1) T. Ando investigated the Hadamard product as a linear operator on the matrix space, and gave a characterization of its norm with respect to the numerical radius. He succeeded in complete parametrization of all extreme points of some convex set of matrices. On the basis of the idea of majorization he established inequalities complementary to the Golden-Thompson inequalities.(2) From the standpoint of matrix analysis H. Schneider obtained a condition for balancing of a weighted direct graphs. He also established a relation between heights and indices for O-pattern graphs.(3) By a method of combinatorics R. Brualdi succeeded in enumerating the number of self-dual codes. He also obtained a basic result for that all matrices with a common graph are invertible.(4) N. Nagai established a method of construction of digital filters by successive extraction of 2-wire lines.(5) T. Nakazi established a relationship between an extremal problem in the Hardy spaces and a property of a Hankel operator, a generalization of a Hankel matrix.(6) Y. Nakamura extended some extremal problem for matrices in terms of majorization to a problem for a metric in function space and analyzed the structure of its solutions. he also established a formula of norm for a special product of matrices.