A Journey into Matrix Analysis

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.