Ch. 1. Some fundamental notions. 1.1. Definitions. 1.2. Components of a matrix. 1.3. Matrix functions. 1.4. Normal matrices -- ch. 2. Evolving systems -- ch. 3. Markov chains. 3.1. Non-negative matrices. 3.2. General properties -- ch. 4. Glass transition -- ch. 5. The Kerner model. 5.1. A simple example: Se-As glass -- ch. 6. Formal developments. 6.1. Spectral aspects. 6.2. Reducibility and regularity. 6.3. Projectors and asymptotics. 6.4. Continuum time -- ch. 7. Equilibrium, dissipation and ergodicity. 7.1. Recurrence, transience and periodicity. 7.2. Detailed balancing and reversibility. 7.3. Ergodicity -- ch. 8. Prelude -- ch. 9. Definition and main properties. 9.1. Bases. 9.2. Double Fourier transform. 9.3. Random walks -- ch. 10. Discrete quantum mechanics. 10.1. Introduction. 10.2. Weyl-Heisenberg groups. 10.3. Weyl-Wigner transformations. 10.4. Braiding and quantum groups -- ch. 11. Quantum symplectic structure. 11.1. Matrix differential geometry. 11.2. The symplectic form. 11.3. The quantum fabric -- ch. 12. An organizing tool -- ch. 13. Bell polynomials. 13.1. Definition and elementary properties. 13.2. The matrix representation. 13.3. The Lagrange inversion formula. 13.4. Developments -- ch. 14. Determinants and traces. 14.1. Introduction. 14.2. Symmetric functions. 14.3. Polynomials. 14.4. Characteristic polynomials. 14.5. Lie algebras invariants -- ch. 15. Projectors and iterates. 15.1. Projectors, revisited. 15.2. Continuous iterates -- ch. 16. Gases: real and ideal. 16.1. Microcanonical ensemble. 16.2. The canonical ensemble. 16.3. The grand canonical ensemble. 16.4. Braid statistics. 16.5. Condensation theories. 16.6. The Fredholm formalism.

The First Part Of This Book Begins With An Introduction To Matrices Through Linear Transformations On Vector Spaces, Followed By A Discussion On The Algebra Of Matrices, Special Matrices, Linear Equations, The Eigenvalue Problem, Bilinear And Quadratic Forms, Kronecker Sum And Product Of Matrices. Other Matrices Which Occur In Physics, Such As The Rotation Matrix, Pauli Spin Matrices And Dirac Matrices, Are Then Presented. A Brief Account Of Infinite Matrices From The Point Of View Of Matrix Formulation Of Quantum Mechanics Is Also Included. The Emphasis In This Part Is On Linear Dependence And Independence Of Vectors And Matrices, Linear Combinations, Independent Parameters Of Various Special Matrices And Such Other Concepts As Help The Student In Obtaining A Clear Understanding Of The Subject. A Simplified Proof Of The Theorem That A Common Set Of Eigenvectors Can Be Found For Two Commuting Matrices Is Given. The Second Part Deals With Cartesian And General Tensors. Many Physical Situations Are Discussed Which Require The Use Of Second And Higher Rank Tensors, Such As Effective Mass Tensor, Moment Of Inertia Tensor, Stress, Strain And Elastic Constants, Piezoelectric Strain Coefficient Tensor, Etc. Einsteins Summation Convention Is Explained In Detail And Common Errors Arising In Its Use Are Pointed Out. Rules For Checking The Correctness Of Tensor Equations Are Given. This Is Followed By Four-Vectors In Special Relativity And Covarient Formulation Of Electrodynamics. This Part Comes To An End With The Concept Of Parallel Displacement Of Vectors In Riemannian Space And Covariant Derivative Of Tensors, Leading To The Curvature Tensors And Its Properties.Appendix I Has Expanded And Two New Appendices Have Been Added In This Edition.

This is an updated translation of a book published in Czech by the SNTL - Publishers of Technical Literature in 1981. In developing this book, it was found reasonable to consider special matrices in general sense and also to include some more or less auxiliary topics that made it possible to present some facts or processes more demonstratively. An example is the graph theory. Chapter 1 contains the definitions of basic concepts of the theory of matrices, and fundamental theorems. The Schur complement is defined here in full generality and using its properties we prove the theorem on the factorization of a partitioned matrix into the product of a lower block triangular matrix with identity diagonal blocks, a block diagonal matrix, and an upper block triangular matrix with identity diagonal blocks. The theorem on the Jordan normal form of a matrix is gi¥en without proof. Chapter 2 is concerned with symmetric and Hermitian matrices. We prove Schur's theorem and, using it, we establish the fundamental theorem describing the factorization of symmetric or Hermitian matrices. Further, the properties of positive definite and positive semidefinite matrices are studied. In the conclusion, Sylvester's law of inertia of quadratic forms and theorems on the singular value decomposition and polar decomposition are proved. Chapter 3 treats the mutual connections between graphs and matrices.

This revised and corrected second edition of a classic on special matrices provides researchers in numerical linear algebra and students of general computational mathematics with an essential reference. 1986 edition.

The theory of vector spaces and matrices is an essential part of the mathematical background required by physicists. This book is written primarily as a text for the undergraduate and postgraduate students and as a reference for physicists. Special emphasis is given to topics relevant to physics, e.g., linear independence and dependence of vectors, inner product, orthonormality, matrices as representations of linear transformations on vector spaces, similarity, eigenvalues, eigenvectors and diagonalization of matrices etc. The role of orthogonal, Hermitian and unitary matrices in physics is highlighted. A large number of solved problems and exercises, with enough hints/solutions, are provided to make the book self sufficient.

Mathematics is an essential ingredient in the education of a student of mathematics or physics of a professional physicist, indeed in the education of any professional scientist or engineer. The purpose of Mathematical Physics is to provide a comprehensive study of the mathematics underlying theoretical physics at the level of graduate and postgraduate students and also have enough depth for others interested in higher level mathematics relevant to specialized fields. It is also intended to serve the research scientist or engineer who needs a quick refresher course in the subject. The Fourth Edition of the book has been thoroughly revised and updated keeping in mind the requirements of students and the latest UGC syllabus.

This book focuses on the unifying power of the geometrical language in bringing together concepts from many different areas of physics, ranging from classical physics to the theories describing the four fundamental interactions of Nature -- gravitational, electromagnetic, strong nuclear, and weak nuclear. The book provides in a single volume a thorough introduction to topology and differential geometry, as well as many applications to both mathematical and physical problems. It is aimed as an elementary text and is intended for first year graduate students. In addition to the traditional contents of books on special and general relativities, this book discusses also some recent advances such as de Sitter invariant special relativity, teleparallel gravity and their implications in cosmology for those wishing to reach a higher level of understanding.