1st Edition

# Linear Algebra to Differential Equations

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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 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.

**1. Vectors and Matrices.** 1.1. Introduction. 1.2. Scalars and Vectors. 1.3. Introduction to Matrices. 1.4. Types of Matrices. 1.5. Elementary Operations and Elementary Matrices. 1.6. Determinants. 1.7. Inverse of a Matrix. 1.8. Partitioning of Matrices. 1.9. Advanced Topics: Pseudo Inverse and Congruent Inverse. 1.10. Conclusion. **2. Linear System of Equations.** 2.1. Introduction. 2.2. Linear System of Equations. 2.3. Rank of a Matrix. 2.4. Echelon Form and Normal Form. 2.4. Echelon Form and Normal Form. 2.5. Solutions of a Linear System of Equations. 2.6. Cayley Hamilton Theorem. 2.7. Eigen Values and Eigen Vectors. 2.8. Singular Values and Singular Vectors. 2.9. Quadratic Forms. 2.10. Conclusion. **3. Vector Spaces.** 3.1. Introduction. 3.2. Vector Space and a Subspace. 3.3. Linear Independence, Basis and Dimension. 3.4. Change of Basis – Matrix. 3.5. Linear Transformations. 3.6. Matrices of Linear Transformations. 3.7. Inner Product Space. 3.8. Gram-Schmidt Orthogonalization. 3.9. Linking Linear Algebra to Differential Equations. 3.10. Conclusion. **4. Numerical Methods in Linear Algebra.** 4.1. Introduction. 4.2. Elements of Computation and Errors. 4.3. Direct Methods for Solving a Linear System of Equations. 4.4. Iterative Methods. 4.5. Householder Transformation. 4.6. Tridiagonalization of a Symmetric Matrix by Plane Rotation. 4.7. QR Decomposition. 4.8. Eigen Values: Bounds and Power Method. 4.9. Krylov Subspace Methods. 4.10. Conclusion.** 5. Applications.** 5.1. Introduction. 5.2. Finding Curves through Given Points. 5.3. Markov Chains. 5.4. Leontief's Models. 5.5. Cryptology. 5.6. Application to Computer Graphics. 5.7. Application to Computer Graphics. 5.8. Bioinformatics. 5.9. Principal Component Analysis (PCA). 5.10. Big Data. 5.11. Conclusion. **6. Kronecker Product.** 6.1. Introduction. 6.2. Primary Matrices. 6.3. Kronecker Products. 6.4. Further Properties of Kronecker products. 6.5. Kronecker product of two linear transformations. 6.6. Kronecker Product and Vector Operators. 6.7. Permutation Matrices and Kronecker Products. 6.8. Analytical functions and Kronecker Product. 6.9. Kronecker Sum. 6.10. Lyapunov Function. 6.11. Conclusion. **7. Calculus of Matrices. **7.1. Introduction. 7.2. Derivative of a Matrix Valued Function with respect to a Scalar. 7.3. Derivative of a Vector Valued Function w.r.t. a Vector. 7.4. Derivative of a Scalar Valued Function w.r.t. a Matrix. 7.5. Derivative of a Matrix Valued Function w.r.t. its Entries and Vice versa. 7.6. The Matrix Differential. 7.7. Derivative of a Matrix w.r.t. a Matrix. 7.8. Derivative Formula using Kronecker Products. 7.9. Another Definition for Derivative of a Matrix w.r.t. a Matrix. 7.10. Conclusion. **8. Linear Systems of Differential Equations.** 8.1. Introduction. 8.2. Linear Systems. 8.3. Fundamental Matrix. 8.4. Method of Successive Approximations. 8.5. Nonhomogeneous Systems. 8.6. Linear Systems with Constant Coefficients. 8.7. Stability Analysis of a System. 8.8. Election Mathematics. 8.9. Conclusion. **9. Linear Matrix Differential Equations.** 9.1. Introduction. 9.2. Initial Value Problem (IVP). 9.3. LMDE X’ = A(t)X. 9.4. The LMDE X’ = AXB. 9.5. More General LMDE. 9.6. A Class of LMDE of Higher Order. 9.7. Boundary Value Problem of LMDE. 9.8. Trigonometric and Hyperbolic Matrix Functions. 9.9. Conclusion.

### Biography

**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 ~~P~~rofessor 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.