This is the Habilitation Thesis manuscript presented at Besan\c{c}on on January 5, focusing on Matrix Analysis, Matrix Inequalities and Matrix Decompositions. There are also some topics in (Hilbert space) Operator Theory. The text should be of interest for a large audience of researchers and students in pure and applied mathematics. We may divide it into five parts: 1) Chapter 1 is an introductory chapter, some results from the period 1999-2010 are given, and a few conjectures are proposed. 2) Chapters 2-4 deal with matrix inequalities, Chapter 2 is concerned with norm inequalities and logmajorization and Chapters 3-4 with functional calculus and a unitary orbit technique that I started to develop in 2003. 3) Chapter 5 is a time-break in infinite dimensional Hilbert space operators, the essential numerical range plays a key role. 4) Chapters 6-8 establish several decompositions for partitioned matrices, especially for positive block matrices. Some norm inequalities involving the numerical range are derived. 5) Chapter 9 may be of special interest for students : a proof of the Spectral Theorem for bounded operators is derived from the matrix case.

翻译：本文为作者于1月5日在贝桑松提交的教授资格论文手稿，聚焦于矩阵分析、矩阵不等式与矩阵分解等主题，同时涉及（希尔伯特空间）算子理论中的若干议题。本稿可供纯数学与应用数学领域的研究者及学生广泛参考。全文可分为五部分：1）第一章为导论章节，梳理了1999-2010年间的部分研究成果，并提出若干猜想；2）第二至四章论述矩阵不等式：第二章探讨范数不等式与对数优超，第三至四章则结合函数演算及作者自2003年开始发展的酉轨道技巧展开分析；3）第五章将视角转向无穷维希尔伯特空间算子，其中本质数值值域起关键作用；4）第六至八章构建了分块矩阵的若干分解方法，特别关注正定分块矩阵，并导出涉及数值值域的若干范数不等式；5）第九章对学子尤具价值：从矩阵情形推导出有界算子的谱定理证明。