Ulteriori informazioni
This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations.
The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several 'MATLAB-Minutes' students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exercises.
Sommario
Linear Algebra in every day life.- Basic mathematical concepts.- Algebraic structures.- Matrices.- The echelon form and the rank of matrices.- Linear systems of equations.- Determinants of matrices.- The characteristic polynomial and eigenvalues of matrices.- Vector spaces.- Linear maps.- Linear forms and bilinear forms.- Euclidean and unitary vector spaces.- Adjoints of linear maps.- Eigenvalues of endomorphisms.- Polynomials and the Fundamental Theorem of Algebra.- Cyclic subspaces, duality and the Jordan canonical form.- Matrix functions and systems of differential equations.- Special classes of endomorphisms.- The singular value decomposition.- The Kronecker product and linear matrix equations.
Info autore
Jörg Liesen's research interests are in Numerical Linear Algebra, Matrix Theory and Constructive Approximation, with a particular focus on the convergence and stability analysis of iterative methods. He is also interested in the history of Mathematics, and in particular of Linear Algebra. He is the recipient of several prizes and awards for his mathematical work, including the Householder Award, the Emmy Noether Fellowship and the Heisenberg Professorship of the DFG. He likes to teach and pursue Mathematics as a lively subject, connecting theory with an ever increasing variety of fascinating applications.
Volker Mehrmann's research interests are in Numerical Mathematics, Control Theory, Matrix Theory as well as Scientific Computing. In recent years he has focused on the development and analysis of numerical methods for nonlinear eigenvalue problems and differential-algebraic systems with applications in many fields such as mechanical systems, electronic circuit simulation and acoustic field computations. He is co-editor-in-chief of the journal Linear Algebra and its Applications and editor of many other journals in Linear Algebra and Numerical Analysis. He believes that Mathematics has become a central prerequisite for the societal development of the 21st century and that mathematical methods play the key role in the modeling, simulation, control and optimization of all are as of technological development.
Riassunto
This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations.
The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exercises.
Relazione
"This book begins with applied problems that are examined as the theory is developed. ... Liesen and Mehrmann ... present computations with matrix groups and rings, elementary matrices, echelon forms, rank, linear systems, determinants, and eigenvalues and eigenvectors before introducing vectors and vector spaces. ... Each chapter ends with a set of exercises addressing both computation and theory. ... Summing Up: Recommended. Upper-division undergraduates through professionals/practitioners." (J. R. Burke, Choice, Vol. 53 (12), September, 2016)
"It provides a good introductory undergraduate course at an intermediate level ... . From the beginning, the authors motivate the material with interesting examples ... . the text includes short 'MATLAB-Minutes' which are exercises providing an informal introduction to the use of MATLAB in linear algebra, including hints about how care may be needed when working in finite precision." (John D. Dixon, zbMATH 1334.15001, 2016)
