#### Objective

Course Main Goal
The main goal of this course is to concisely present fundamental ideas, results and techniques in matrix theory.
Course Description
The first part of the course deals with the partitioned matrices with some applications, the continuity argument principle and the Singular Value Decomposition theorem (SVD). Then the second part is concerned with the unitary matrices and contractions. The third part studies the positive semidefinite matrices and Hermitian matrices. The operator monotone and operator convex functions with some characterisations and the Lowner’s theorem is studied in the fourth part. The last part of this course deals with the norm inequalities such as: Lowner-Heinz inequalities, inequalities for matrix powers, arithmetic-geometric mean inequalities, inequalities of Holder and Minkowski types, the grand Furuta inequalities, and trace inequalities and some conjectures.

#### Content

Chapter 1: Partitioned Matrices
Linear transformations and eigenvalues
Elementary operations of partitioned matrices
The determinant and inverse of partitioned matrices
The inverse of the a sum
The rank of product and sum
The continuity argument
Singular value decomposition theorem (SVD) and polar decomposition
Majorisations and eigenvalues
Chapter 2: Unitary Matrices and Contractions
Properties of unitary matrices
Metric spaces and contractions
Contractions and unitary matrices
A trace inequalities of unitary matrices
Chapter 3: Positive Semidefinite Matrices
Positive semidefinite matrices
A pair of positive semidefinte matrices
Partitioned positive semidefinte matrices
Schur complements and determinantal inequalities
Schur complements and Hadamard product
The Cauchy-Schwarz and Kantorovich inequalities
Hermitian matrices
The product of hermitian matrices
The Min-Max theorem and interlacing theorem
Eigenvalues and singular values inequalities
A triangle inequality for the matrix (A*A)1/2
Chapter 4: Operator Monotone and Operator Convex Functions
Definitions and simple examples
Inequalities in the Lowner partial order
Some characterisations
Smoothness properties
Lowner’s theorem
Chapter 5: Norm Inequalities
Cartesian decomposition revisited
Arithmetic-Geometric mean inequalities
Lowner-Heinz inequalities
Inequalities of Holder and Minkowski types
Inequalities for the exponential function and the Golden-Thompson inequality
The Furuta’s inequalities
Some trace inequalities
Open problems