Advanced Matrix Theory Description

Key Elements


MDIS 502


M2 Discrete Mathematics and Algebra





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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.


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