Contents Chapter 1 Basics of Linear Algebra 1 1.1 Basic operations of matrices 2 1.1.1 Addition for matrices 3 1.1.2 Scalar multiplication of matrix A 4 1.1.3 Matrix multiplication 5 1.1.4 Identity matrix 6 1.1.5 Transposition matrix and conjugate transpose of matrix A 1.1.6 Matrix inversion 7 1.1.7 Symmetries 8 1.2 Determinants 11 1.2.1 Determinant definition 11 1.2.2 Properties of determinants 12 1.2.3 Cofactor 14 1.2.4 Cramer’s rule 18 1.3 Elementary operations 21 1.3.1 Elementary row transformation 21 1.3.2 Reduced echelon form 23 1.3.3 Rank of matrix 25 1.3.4 Solving equations by elementary transformation 26 1.4 Linear independence 30 1.5 Exercises 35 Chapter 2 Linear Space 40 2.1 Set and map 40 2.2 Linear space 41 2.3 Basis, dimension and coordinates 47 2.4 Change of basis 58 2.5 Exercises 62 Chapter 3 Normed Linear Space and Inner Product Space 66 3.1 Normed linear space and matrix norm 66 3.1.1 Normed linear space 66 3.1.2 Norm of matrix 68 3.2 Inner product spaces 71 3.2.1 Inner product 71 3.2.2 Representation of inner product 77 3.2.3 Orthogonality and Schmidt’s orthogonalization method 79 3.3 Application of norm-preliminary matrix analysis 88 3.3.1 The limit of matrix sequence 88 3.3.2 Matrix series 90 3.3.3 Matrix power series 92 3.3.4 Differentiation and integration of matrices 95 3.4 Exercises 96 Chapter 4 Linear Transformation 101 4.1 Linear transformation 101 4.2 Matrix of linear transformation .109 4.3 Eigenvalues and eigenvectors .119 4.4 Eigenvalues and eigenvectors for matrix 22 4.5 Exercises 26 Chapter 5 Jordan Normal Form of Matrix and Matrix Function 31 5.1 Diagonalization 131 5.2 Jordan normal form of matrix A 35 5.3 Minimum polynomial 146 5.4 Matrix functions 150 5.4.1 Matrix function by infinite series 50 5.4.2 General definition and calculation of matrix function 155 5.4.3 Applications 159 5.5 Exercises 63 Chapter 6 Applications of Matrix Theory in Linear Equations and Matrix Equations 66 6.1 Matrix factorization and application in linear equations 166 6.1.1 The LU factorization and applications 166 6.1.2 Applications in solving linear equations 172 6.2 Minus inverse and applications in compatible linear equations 174 6.3 Plus inverse of matrix and the minimal norm least square solutions of linear equations 182 6.3.1 Plus inverse of matrix 182 6.3.2 Plus inverse of matrix 185 6.3.3 Minimal norm least square solution of linear equations 187 6.4 Kronecker product and applications in matrix equations 189 6.4.1 Definitions and properties of Kronecker product 190 6.4.2 Applications in matrix equations 193 6.5 Exercises 196 References 198
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