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E-grāmata: Linear Algebra to Differential Equations

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  • Formāts: 412 pages
  • Izdošanas datums: 26-Sep-2021
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781351014946
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Linear Algebra to Differential EquationsPalielināt
  • Formāts: 412 pages
  • Izdošanas datums: 26-Sep-2021
  • Izdevniecība: CRC Press Inc
  • ISBN-13: 9781351014946
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"Linear Algebra to Differential Equations concentrates on the essential topics necessary for all engineering students in general and computer science branch students, in particular. Specifically, the topics dealt will help the reader in applying linear algebra as a tool. The advent of high-speed computers has paved the way for studying large systems of linear equations as well as large systems of linear differential equations. Along with the standard numerical methods, methods that curb the progress of error are given for solving linear systems of equations. The topics of linear algebra and differential equations are linked by Kronecker products and calculus of matrices. These topics are useful in dealing with linear systems of differential equations andmatrix differential equations. Differential equations are treated in terms of vector and matrix differential systems, as they naturally arise while formulating practical problems. The essential concepts dealing with the solutions and their stability are briefly presented to motivate the reader towards further investigation. This book caters to the needs of Engineering students in general and in particular, to students of Computer Science & Engineering, Artificial Intelligence, Machine Learning and Robotics. Further, the book provides a quick and complete overview of linear algebra and introduces linear differential systems, serving the basic requirements of scientists and researchers in applied fields. Features Provides complete basic knowledge of the subject Exposes the necessary topics lucidly Introduces the abstraction and at the same time is down to earth Highlights numerical methods and approaches that are more useful Essential techniques like SVD and PCA are given Applications (both classical and novel) bring out similarities in various disciplines: Illustrative examples for every concept: A brief overview of techniques that hopefully serves the present and future needs of students and scientists"--

The topics of linear algebra and DE are linked by Kronecker products and calculus of matrices. These topics are useful in dealing with linear systems of DE and MDE. This book concentrates on the essential topics necessary for all engineering students in general and computer science branch students, in particular.



Linear Algebra to Differential Equations concentrates on the essential topics necessary for all engineering students in general and computer science branch students, in particular. Specifically, the topics dealt will help the reader in applying linear algebra as a tool.

The advent of high-speed computers has paved the way for studying large systems of linear equations as well as large systems of linear differential equations. Along with the standard numerical methods, methods that curb the progress of error are given for solving linear systems of equations.

The topics of linear algebra and differential equations are linked by Kronecker products and calculus of matrices. These topics are useful in dealing with linear systems of differential equations and matrix differential equations. Differential equations are treated in terms of vector and matrix differential systems, as they naturally arise while formulating practical problems. The essential concepts dealing with the solutions and their stability are briefly presented to motivate the reader towards further investigation.

This book caters to the needs of Engineering students in general and in particular, to students of Computer Science & Engineering, Artificial Intelligence, Machine Learning and Robotics. Further, the book provides a quick and complete overview of linear algebra and introduces linear differential systems, serving the basic requirements of scientists and researchers in applied fields.

Features

  • Provides complete basic knowledge of the subject

  • Exposes the necessary topics lucidly

  • Introduces the abstraction and at the same time is down to earth

  • Highlights numerical methods and approaches that are more useful

  • Essential techniques like SVD and PCA are given

  • Applications (both classical and novel) bring out similarities in various disciplines:

  • Illustrative examples for every concept: A brief overview of techniques that hopefully serves the present and future needs of students and scientists.

Preface xi
1 Vectors and Matrices
1(52)
1.1 Introduction
1(1)
1.2 Scalars and Vectors
1(9)
1.3 Introduction to Matrices
10(10)
1.4 Types of Matrices
20(6)
1.5 Elementary Operations and Elementary Matrices
26(3)
1.6 Determinants
29(5)
1.7 Inverse of a Matrix
34(9)
1.8 Partitioning of Matrices
43(4)
1.9 Advanced Topics: Pseudo Inverse and Congruent Inverse
47(4)
1.10 Conclusion
51(2)
2 Linear System of Equations
53(44)
2.1 Introduction
53(1)
2.2 Linear System of Equations
53(3)
2.3 Rank of a Matrix
56(3)
2.4 Echelon Form and Normal Form
59(6)
2.5 Solutions of a Linear System of Equations
65(7)
2.6 Cayley-Hamilton Theorem
72(3)
2.7 Eigen-values and Eigen-vectors
75(13)
2.8 Singular Values and Singular Vectors
88(2)
2.9 Quadratic Forms
90(5)
2.10 Conclusion
95(2)
3 Vector Spaces
97(34)
3.1 Introduction
97(1)
3.2 Vector Space and Subspaces
97(4)
3.3 Linear Independence, Basis and Dimension
101(5)
3.4 Change of Basis-Matrix
106(3)
3.5 Linear Transformations
109(6)
3.6 Matrices of Linear Transformations
115(6)
3.7 Inner Product Space
121(4)
3.8 Gram-Schmidt Orthogonalization
125(2)
3.9 Linking Linear Algebra to Differential Equations
127(2)
3.10 Conclusion
129(2)
4 Numerical Methods in Linear Algebra
131(50)
4.1 Introduction
131(1)
4.2 Elements of Computation and Errors
131(5)
4.3 Direct Methods for Solving a Linear System of Equations
136(13)
4.4 Iterative Methods
149(11)
4.5 Householder Transformation
160(7)
4.6 Tridiagonalization of a Symmetric Matrix by Plane Rotation
167(3)
4.7 QR Decomposition
170(2)
4.8 Eigen-values: Bounds and Power Method
172(6)
4.9 Krylov Subspace Methods
178(2)
4.10 Conclusion
180(1)
5 Applications
181(44)
5.1 Introduction
181(1)
5.2 Finding Curves through Given Points
181(2)
5.3 Markov Chains
183(7)
5.4 Leontief's Models
190(6)
5.5 Cryptology
196(3)
5.6 Application to Computer Graphics
199(5)
5.7 Application to Robotics
204(7)
5.8 Bioinformatics
211(4)
5.9 Principal Component Analysis (PCA)
215(5)
5.10 Big Data
220(3)
5.11 Conclusion
223(2)
6 Kronecker Product
225(48)
6.1 Introduction
225(1)
6.2 Primary Matrices
225(8)
6.3 Kronecker Products
233(5)
6.4 Further Properties of Kronecker Products
238(7)
6.5 Kronecker Product of Two Linear Transformations
245(2)
6.6 Kronecker Product and Vector Operators
247(4)
6.7 Permutation Matrices and Kronecker Products
251(5)
6.8 Analytical Functions and Kronecker Product
256(5)
6.9 Kronecker Sum
261(7)
6.10 Lyapunov Function
268(3)
6.11 Conclusion
271(2)
7 Calculus of Matrices
273(38)
7.1 Introduction
273(1)
7.2 Derivative of a Matrix Valued Function with Respect to a Scalar
273(7)
7.3 Derivative of a Vector-Valued Function w.r.t. a Vector
280(5)
7.4 Derivative of a Scalar-Valued Function w.r.t. a Matrix
285(3)
7.5 Derivative of a Matrix Valued Function w.r.t. Its Entries and Vice versa
288(7)
7.6 The Matrix Differential
295(2)
7.7 Derivative of a Matrix w.r.t. a Matrix
297(4)
7.8 Derivative Formula using Kronecker Products
301(7)
7.9 Another Definition for Derivative of a Matrix w.r.t. a Matrix
308(1)
7.10 Conclusion
309(2)
8 Linear Systems of Differential Equations
311(36)
8.1 Introduction
311(1)
8.2 Linear Systems
312(3)
8.3 Fundamental Matrix
315(5)
8.4 Method of Successive Approximations
320(4)
8.5 Nonhomogeneous Systems
324(4)
8.6 Linear Systems with Constant Coefficients
328(9)
8.7 Stability Analysis of a System
337(5)
8.8 Election Mathematics
342(2)
8.9 Conclusion
344(3)
9 Linear Matrix Differential Equations
347(30)
9.1 Introduction
347(1)
9.2 Initial-value-problems of LMDEs
347(2)
9.3 LMDEX' = A(t)X
349(3)
9.4 The LMDE X' = AXB
352(3)
9.5 More General LMDE
355(5)
9.6 A Class of LMDE of Higher Order
360(4)
9.7 Boundary Value Problem of LMDE
364(3)
9.8 Trigonometric and Hyperbolic Matrix Functions
367(7)
9.9 Conclusion
374(3)
Bibliography 377(2)
Answers 379(16)
Index 395
Dr. I. Vasundhara Devi is a Professor in Department of Mathematics, Gayatri Vidya Parishad College of Engineering, Visakhapatnam, India and is also Associate Director of Gayatri Vidya Parishad-Professor V. Lakshmikantham Institute for Advanced Studies, Visakhapatnam, India. She is a co-author for a couple of research monographs, edited a couple of journal volumes and published around a hundred peer reviewed research articles.

Dr. Sadashiv G. Deo (S.G.Deo) has an illustrious career both as a Professor and as an author. He retired as Director of Gayatri Vidya Parishad-Professor V. Lakshmikantham Institute for Advanced Studies, Visakhapatnam, India. He has over 35 years of teaching experience and has taught at Mumbai, Marathwada and Goa Universities in India; Texas University at Arlington (TX), Florida Institute of technology, Melbourne (Fl), USA and gave lectures in many well-known Institutions in USA, Canada and India. He co-authored many textbooks, research monographs and lecture notes, of which his text book on Differential equations, is being used as a text book in various countries. He published popular articles in Mathematics in English as well as in Marathi. He published several research papers in peer reviewed journals.

Dr Ramakrishna Khandeparkar worked as Professor of Mathematics at Don Bosco College of Engineering, Fatorda, Goa. He taught for more than 35 years and published many research and expository articles in both national and international peer reviewed journals.
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