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Linear Algebra

作者
：
Arak M. Mathai (著) ; Hans J. Haubold (著)
出版社
：
De Gruyter
出版日期
：
2017/10/23
閱讀格式
：
EPUB
書籍分類
：
人文
學科分類
：
語言文學類
ISBN
：
9783110562590
eigenvalue matrix eigenvalues eigenvector correlation Lagrangian multiplier

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In order not to intimidate students by a too abstract approach, this textbook on linear algebra is written to be easy to digest by non-mathematicians. It introduces the concepts of vector spaces and mappings between them without dwelling on statements such as theorems and proofs too much. It is also designed to be self-contained, so no other material is required for an understanding of the topics covered. As the basis for courses on space and atmospheric science, remote sensing, geographic information systems, meteorology, climate and satellite communications at UN-affiliated regional centers, various applications of the formal theory are discussed as well. These include differential equations, statistics, optimization and some engineering-motivated problems in physics. ContentsVectorsMatricesDeterminantsEigenvalues and eigenvectorsSome applications of matrices and determinantsMatrix series and additional properties of matrices
  • Cover
  • Title Page
  • Copyright
  • Preface
  • Acknowledgement
  • Contents
  • List of Symbols
  • 1 Vectors
    • 1.0 Introduction
    • 1.1 Vectors as ordered sets
    • 1.2 Geometry of vectors
      • 1.2.1 Geometry of scalar multiplication
      • 1.2.2 Geometry of addition of vectors
      • 1.2.3 A coordinate-free definition of vectors
      • 1.2.4 Geometry of dot products
      • 1.2.5 Cauchy–Schwartz inequality
      • 1.2.6 Orthogonal and orthonormal vectors
      • 1.2.7 Projections
      • 1.2.8 Work done
    • 1.3 Linear dependence and linear independence of vectors
      • 1.3.1 A vector subspace
      • 1.3.2 Gram–Schmidt orthogonalization process
    • 1.4 Some applications
      • 1.4.1 Partial differential operators
      • 1.4.2 Maxima/minima of a scalar function of many real scalar variables
      • 1.4.3 Derivatives of linear and quadratic forms
      • 1.4.4 Model building
  • 2 Matrices
    • 2.0 Introduction
    • 2.1 Various definitions
      • 2.1.1 Some more practical situations
    • 2.2 More properties of matrices
      • 2.2.1 Some more practical situations
      • 2.2.2 Pre and post multiplications by diagonal matrices
    • 2.3 Elementary matrices and elementary operations
      • 2.3.1 Premultiplication of a matrix by elementary matrices
      • 2.3.2 Reduction of a square matrix into a diagonal form
      • 2.3.3 Solving a system of linear equations
    • 2.4 Inverse, linear independence and ranks
      • 2.4.1 Inverse of a matrix by elementary operations
      • 2.4.2 Checking linear independence through elementary operations
    • 2.5 Row and column subspaces and null spaces
      • 2.5.1 The row and column subspaces
      • 2.5.2 Consistency of a system of linear equations
    • 2.6 Permutations and elementary operations on the right
      • 2.6.1 Permutations
      • 2.6.2 Postmultiplications by elementary matrices
      • 2.6.3 Reduction of quadratic forms to their canonical forms
      • 2.6.4 Rotations
      • 2.6.5 Linear transformations
      • 2.6.6 Orthogonal bases for a vector subspace
      • 2.6.7 A vector subspace, a more general definition
      • 2.6.8 A linear transformation, a more general definition
    • 2.7 Partitioning of matrices
      • 2.7.1 Partitioning and products
      • 2.7.2 Partitioning of quadratic forms
      • 2.7.3 Partitioning of bilinear forms
      • 2.7.4 Inverses of partitioned matrices
      • 2.7.5 Regression analysis
      • 2.7.6 Design of experiments
  • 3 Determinants
    • 3.0 Introduction
    • 3.1 Definition of the determinant of a square matrix
      • 3.1.1 Some general properties
      • 3.1.2 A mechanical way of evaluating a 3 × 3 determinant
      • 3.1.3 Diagonal and triangular block matrices
    • 3.2 Cofactor expansions
      • 3.2.1 Cofactors and minors
      • 3.2.2 Inverse of a matrix in terms of the cofactor matrix
      • 3.2.3 A matrix differential operator
      • 3.2.4 Products and square roots
      • 3.2.5 Cramer’s rule for solving systems of linear equations
    • 3.3 Some practical situations
      • 3.3.1 Cross product
      • 3.3.2 Areas and volumes
      • 3.3.3 Jacobians of transformations
      • 3.3.4 Functions of matrix argument
      • 3.3.5 Partitioned determinants and multiple correlation coefficient
      • 3.3.6 Maxima/minima problems
  • 4 Eigenvalues and eigenvectors
    • 4.0 Introduction
    • 4.1 Eigenvalues of special matrices
    • 4.2 Eigenvectors
      • 4.2.1 Some definitions and examples
      • 4.2.2 Eigenvalues of powers of a matrix
      • 4.2.3 Eigenvalues and eigenvectors of real symmetric matrices
    • 4.3 Some properties of complex numbers and matrices in the complex fields
      • 4.3.1 Complex numbers
      • 4.3.2 Geometry of complex numbers
      • 4.3.3 Algebra of complex numbers
      • 4.3.4 n-th roots of unity
      • 4.3.5 Vectors with complex elements
      • 4.3.6 Matrices with complex elements
    • 4.4 More properties of matrices in the complex field
      • 4.4.1 Eigenvalues of symmetric and Hermitian matrices
      • 4.4.2 Definiteness of matrices
      • 4.4.3 Commutative matrices
  • 5 Some applications of matrices and determinants
    • 5.0 Introduction
    • 5.1 Difference and differential equations
      • 5.1.1 Fibonacci sequence and difference equations
      • 5.1.2 Population growth
      • 5.1.3 Differential equations and their solutions
    • 5.2 Jacobians of matrix transformations and functions of matrix argument
      • 5.2.1 Jacobians of matrix transformations
      • 5.2.2 Functions of matrix argument
    • 5.3 Some topics from statistics
      • 5.3.1 Principal components analysis
      • 5.3.2 Regression analysis and model building
      • 5.3.3 Design type models
      • 5.3.4 Canonical correlation analysis
    • 5.4 Probability measures and Markov processes
      • 5.4.1 Invariance of probability measures
      • 5.4.2 Discrete time Markov processes and transition probabilities
    • 5.5 Maxima/minima problems
      • 5.5.1 Taylor series
      • 5.5.2 Optimization of quadratic forms
      • 5.5.3 Optimization of a quadratic form with quadratic form constraints
      • 5.5.4 Optimization of a quadratic form with linear constraints
      • 5.5.5 Optimization of bilinear forms with quadratic constraints
    • 5.6 Linear programming and nonlinear least squares
      • 5.6.1 The simplex method
      • 5.6.2 Nonlinear least squares
      • 5.6.3 Marquardt’s method
      • 5.6.4 Mathai–Katiyar procedure
    • 5.7 A list of some more problems from physical, engineering and social sciences
      • 5.7.1 Turbulent flow of a viscous fluid
      • 5.7.2 Compressible flow of viscous fluids
      • 5.7.3 Heat loss in a steel rod
      • 5.7.4 Small oscillations
      • 5.7.5 Input–output analysis
  • 6 Matrix series and additional properties of matrices
    • 6.0 Introduction
    • 6.1 Matrix polynomials
      • 6.1.1 Lagrange interpolating polynomial
      • 6.1.2 A spectral decomposition of a matrix
      • 6.1.3 An application in statistics
    • 6.2 Matrix sequences and matrix series
      • 6.2.1 Matrix sequences
      • 6.2.2 Matrix series
      • 6.2.3 Matrix hypergeometric series
      • 6.2.4 The norm of a matrix
      • 6.2.5 Compatible norms
      • 6.2.6 Matrix power series and rate of convergence
      • 6.2.7 An application in statistics
    • 6.3 Singular value decomposition of a matrix
      • 6.3.1 A singular value decomposition
      • 6.3.2 Canonical form of a bilinear form
  • References
  • Index
  • 出版地 ： 德國
  • 語言 ： 德文
Linear Algebra

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