Matrix Analysis and Its Applications to Signal Processing

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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.