# Modern Mathematical Methods In Technology

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Modern Mathematical Methods in Technology deals with applied mathematics and its finite methods. The book explains the linear algebra, optimization theory, and elements of the theory of graphs. This book explains the matrix theory and analysis, as well as the applications of matrix calculus. It discusses the linear mappings, basic matrix operations, hypermatrices, vector systems, and other algebraic concepts. In addition, it presents the sequences, series, continuity, differentiation, and integration of matrices, as well as the analytical matrix functions. The book discusses linear optimization, linear programming problems, and their solution. It also describes transportation problems and their solution by Hungarian method, as well as convex optimization and the Kuhn-Tucker theorem. The book discusses graphs including sub-, complete, and complementary graphs. It also presents the Boolean algebra and Ford-Fulkerson theorem. This book is invaluable to Math practitioners and non-practitioners.

## Table of Contents

Editorial Note

Introduction

Chapter 1 Linear Algebra

101. Matrix Theory

101.01. Linear mappings

101.02. Matrices

101.03. Basic matrix operations

101.04. Hypermatrices

101.05. Linearly independent vectors

101.06. Orthogonal and biorthogonal systems of vectors

101.07. The inverse of a matrix

101.08. The dyadic decomposition of matrices

101.09. The rank of a vector system

101.10. The rank of a matrix

101.11. The minimal decomposition of a matrix

101.12. A few theorems on products of matrices

101.13. The dyadic decomposition of certain important matrices

101.14. Eigenvalues and eigenvectors of matrices

101.15. Symmetric and hermitian matrices

101.16. Matrix polynomials

101.17. The characteristic polynomial of a matrix. The Cayley-Hamilton theorem

101.18. The minimum polynomial of a matrix

101.19. The biorthogonal minimal decomposition of a square matrix

102. Matrix Analysis

102.01. Sequences, series, continuity, differentiation and integration of matrices

102.02. Power series of matrices

102.03. Analytical matrix functions

102.04. Decomposition of rational matrices

103. A Few Applications of Matrix Calculus

103.01. The theory of systems of linear equations

103.02. Linear integral equations

103.03. Linear systems of differential equations

103.04. The motion of a particle

103.05. The stability of linear systems

103.06. Bending of a supported beam

103.07. Application of matrix techniques to linear electrical networks

103.08. The application of matrices to the theory of four-pole devices

Chapter 2 Optimization Theory

201. Linear Optimization

201.01. The problem

201.02 Geometrical approaches

201.03. Minimum vectors for a linear programming problem

201.04. Solution of the linear programming problem

201.05. Dual linear programming problems

201.06. Transportation problems and their solution by the Hungarian method

202. Convex Optimization

202.01. The problem

202.02. Definitions and lemmas

202.03. The Kuhn-Tucker theorem

202.04. Convex optimization with differentiable functions

Chapter 3 Elements of the Theory of Graphs

301.01. Introduction

301.02. The idea of a graph

301.03. Sub-graphs and complete graphs; complementary graphs

301.04. Chains, paths and cycles

301.05. Components and blocks of a graph

301.06. Trees and spanning trees of a graph

301.061. An application

301.07. Fundamental systems of cycles and sheaves

301.08. Graphs on surfaces

301.09. Duality

301.10. Boolean algebra

301.101. Incidence matrices

301.102. Cycle matrices

301.103. Sheaf matrices

301.104. Vectors spaces generated by graphs

301.11. Directed graphs

301.111. Matrices associated with directed graphs

301.12. The application of graph theory to the theory of electric networks

301.13. The Ford-Fulkerson theorem

Bibliography

Index

## Product details

- No. of pages: 334
- Language: English
- Copyright: © North Holland 1975
- Published: January 1, 1975
- Imprint: North Holland
- eBook ISBN: 9780444601902

## About the Author

### S. Fenyo

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