J. Vasundhara Devi, Sadashiv G. Deo, Ramakrishna Khandeparkar

Linear Algebra to Differential Equations (eBook, ePUB)

• Format: ePub

Produktbeschreibung

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.

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Produktdetails

• Verlag: Taylor & Francis
• Seitenzahl: 412
• Erscheinungstermin: 26. September 2021
• Englisch
• ISBN-13: 9781351014939
• Artikelnr.: 62409625

Autorenporträt

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.

Inhaltsangabe

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.