Vector Spaces and Matrices in Physics

The theory of vector spaces and matrices is an essential part of the mathematical background required by physicists. Most books on the subject, however, do not adequately meet the requirements of physics courses-they tend to be either highly mathematical or too elementary. Books that focus on mathematical theory may render the subject too dry to hold the interest of physics students, while books that are more elementary tend to neglect some topics that are vital in the development of physical theories. In particular, there is often very little discussion of vector spaces, and many books introduce matrices merely as a computational tool. Vector Spaces and Matrices in Physics fills the gap between the elementary and the heavily mathematical treatments of the subject with an approach and presentation ideal for graduate-level physics students. After building a foundation in vector spaces and matrix algebra, the author takes care to emphasize the role of matrices as representations of linear transformations on vector spaces, a concept of matrix theory that is essential for a proper understanding of quantum mechanics. He includes numerous solved and unsolved problems, and enough hints for the unsolved problems to make the book self-sufficient. Developed through many years of lecture notes, Vector Spaces and Matrices in Physics was written primarily as a graduate and post-graduate textbook and as a reference for physicists. Its clear presentation and concise but thorough coverage, however, make it useful for engineers, chemists, economists, and anyone who needs a background in matrices for application in other areas.

Table of Contents:
ALGEBRAIC SYSTEMS: AN INTRODUCTION Abstract Algebraic Systems Properties of Binary Operations Group Field Ring Functions or Mappings VECTOR SPACES Generalization from Physical to Abstract Vectors Vector Space Subspace Linear Combination of Vectors Linear Independence and Dependence Basis and Dimension: Coordinates Isomorphism of Vector Spaces Inner Product of Vectors Norm (or Length) of a Vector Distance Between Two Vectors Schwarz Inequality Orthogonality LINEAR TRANSFORMATIONS Definition Equality Suma and Scalar Multiple Zero Transformation Idempotent Transformation Nilpotent Transformation Nonsingular Transformation Orthogonal Transformation BASIC MATRIX ALGEBRA AND SPECIAL MATRICES Definition Equality Sum and Difference of Matrices Scalar Multiple of a Matrix Matrix Multiplication Row and Column Vectors Transpose of a Matrix Conjugate of a Matrix Conjugate-Transpose (Hermitian-Conjugate of a Matrix) Trace of a Square Matrix Special Square Matrices Adjoint of a Matrix Determination of Inverse of a Matrix Vector Space of Matrices RANK OF A MATRIX Row and Column Vectors of a Matrix Rank of a Matrix Row Space and Column Space of a Matrix Elementary Row Operations on a Matrix SYSTEMS OF LINEAR EQUATIONS Homogeneous and Non-Homogenous Linear Systems Matrix Form of a Linear System Existence and Uniqueness Theorems A Practical Method of Solving Linear Systems: Gauss Elimination MATRICES AND LINEAR TRANSFORMATIONS Matrix Representation of a Linear Transformation Representation of Product of Transformations Change of Bases and Similarity Transformation EIGENVALUES AND EIGENVECTORS OF A MATRIX Eigenvalues and Eigenvectors Determination of Eigenvalues and Eigenvectors Linear Independence of Eigenvectors Eigenvalues and Eigenvectors of Similar Matrices Eigenvalues of a Diagonal Matrix Hermitian, Skew-Hermitian, and Unitary Matrices Diagonalization of a Matrix Simultaneous Diagonalization and Commutativity An Application: Reduction of Coupled Differential Equations of Matrix Eigenvalue Problem CALEY-HAMILTON THEOREM. MINIMAL POLYNOMIAL OF A MATRIX Caley-Hamilton Theorem Determination of Inverse of a Matrix Minimal Polynomial of a Matrix A Criterion for Diagonalizability FUNCTIONS OF A MATRIX Power of a Matrix Matrix Polynomial Matrix Power Series Evaluation of Matrix Functions for Diagonalizable Matrices Evaluation of Matrix Functions Using Minimal Polynomial BILINEAR, QUADRATIC, HERMITIAN, AND SKEW-HERMITIAN FORMS Bilinear Form Quadratic Form Hermitian (Skew-Hermitian) Form Reduction of a Quadratic Form to Canonical Form Principal Axes Transformation ANSWERS/HINTS TO EXERCISES