Problems and Solutions in Introductory and Advanced Matrix Calculus

This book provides an extensive collection of problems with detailed solutions in introductory and advanced matrix calculus. Supplementary problems in each chapter will challenge and excite the reader, ideal for both graduate and undergraduate mathematics and theoretical physics students. The coverage includes systems of linear equations, linear differential equations, integration and matrices, Kronecker product and vec-operation as well as functions of matrices. Furthermore, specialized topics such as spectral theorem, nonnormal matrices and mutually unbiased bases are included. Many of the problems are related to applications for group theory, Lie algebra theory, wavelets, graph theory and matrix-valued differential forms, benefitting physics and engineering students and researchers alike. It also branches out to problems with tensors and the hyperdeterminant. Computer algebra programs in Maxima and SymbolicC++ have also been provided.

Contents:

• Preface
• Notation
• Basic Operations
• Linear Equations
• Kronecker Product
• Traces, Determinants and Hyperdeterminants
• Eigenvalues and Eigenvectors
• Spectral Theorem
• Commutators and Anticommutators
• Decomposition of Matrices
• Functions of Matrices
• Cayley–Hamilton Theorem
• Hadamard Product
• Norms and Scalar Products
• vec Operator
• Nonnormal Matrices
• Binary Matrices
• Star Product
• Unitary Matrices
• Groups, Lie Groups and Matrices
• Lie Algebras and Matrices
• Braid Group
• Graphs and Matrices
• Hilbert Spaces and Mutually Unbiased Bases
• Linear Differential Equations
• Differentiation and Matrices
• Integration and Matrices
• Bibliography
• Index

Readership: Undergraduate and graduate students and researchers in mathematics, physics and engineering.
Key Features:

• Each chapter starts with an introduction giving definitions and outlining techniques
• Includes modern techniques in matrix calculus
• Considers applications of matrix calculus in group theory and Lie algebra