| As we know, matrix inequality, which is a primary subject of matrix theory, plays an important role in mathematical theory. Not only it has been widely used in various mathematic fields, but also applied in mechanics, control theory, signal processing, communications engineering, systems engineering and other disciplines. It has been discussed actively by academics. In this thesis, the famous inequality of Von Neumann trace and Fischer-type determinantal inequalities for accretive-dissipative matrices was introduced. The main content and achievements are as followings:In part one:The background knowledge about the matrix inequality theory system, including its development history, the previous research, and its application value in technology production was introduced. Also some related basic definitions and lemmas were mentioned.In part two:Mainly researched on the inequality of Von Neumann trace: included the modality of conventional Von Neumann trace inequality. Then, the inequality of Von Neumann trace was studied by using the properties of the matrix divided into blocks, singular value and eigenvalue of the matrix. As a result, the inequalities of the matrix product trace were extended under the certain conditions, and the established conclusions.In part three:Fischer-type determinantal inequalities for accretive-dissipative matrices, which contained the definitions and applications of accretive-dissipative matrices, was discussed. Using the matrix divided into blocks, the Fischer-type determinantal inequalities for accretive-dissipative matrices was defined. Whereafter, inspired by research Kh.D.Ikramov, Lin M and other scholars, we discussed the promotion of the Fischer-type determinantal inequalities for accretive-dissipative matrices. The results which we get are even more accurate than ever, and there will be application in numerical algebra or other fields.In part four:we presented a summary about our work so far and planned the future research. |
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