درس‌هایی درباره ماتریس‌ها

The dynamics of complex systems can clarify the creation of structures in Nature. This creation is driven by the collective interaction of constitutive elements of the system. Such interactions are frequently nonlinear and are directly responsible for the lack of prediction in the evolution process. The self-organization accompanying these processes occurs all around us and is constantly being rediscovered, under the guise of a new jargon, in apparently unrelated disciplines. This volume offers unique perspectives on aspects of fractals and complexity and, through the examination of complementary techniques, provides a unifying thread in this multidisciplinary endeavour. Do nonlinear interactions play a role in the complexity management of socio-economic-political systems? Is it possible to extract the global properties of genetic regulatory networks without knowing the details of individual genes? What can one learn by transplanting the self-organization effects known in laser processes to the study of emotions? What can the change in the level of complexity tell us about the physiological state of the organism? The reader will enjoy finding the answers to these questions and many more in this book. coll17-frnt.pdf -1 Frontmatter 1 Title 1 Preface 2 Contents 3 Corrigenda 6 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chI.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors 8 I. Linear transformations and vectors 8 2. Linear dependence 9 3. Linear vector functions and matrices 10 4. Scalar matrices 12 5. Powers of a matrix; adjoint matrices 13 6. The transverse of a matrix 15 7. Bilinear forms 16 8. Change of basis 16 9. Reciprocal and orthogonal bases 18 10. The rank of a matrix 21 11. Linear dependence 23 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chII.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Characteristic Equation 27 I. Identities 27 2. Matric polynomials in a scalar variable 27 3-4. The division transformation 28 5-6. The characteristic equation 30 7-8. Matrices with distinct roots 32 9-12. Matrices with mulitple roots 34 13. The square root of a matrix 37 14. Reducible matrices 38 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chIII.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors 40 I. Elementary transformations 40 2. The normal form of a matrix 41 3. Determinantal and invariant factors 43 4. Non-singular linear polynomials 44 5. Elementary divisors 45 6-7. Matrices with given elementary divisors 46 8-9. Invariant vectors 50 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chIV.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials 54 I. Vector polynomials 54 2. The degree invariants 55 3-4. Elementary sets 56 5. Linear elementary bases 59 6. Singular linear polynomials 62 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chIX.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form 70 I. Automorphic transformation 70 2-3. The equation y' = +/-aya^-1 71 4. Principal idempotent and nilpotent elements 72 5. The exponential solution 74 6. Matrices which admit a given transformation 75 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chV.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices 77 I. Compound Matrices 77 2. The scalar product 77 3. Compound matrices 78 4. Roots of compound matrices 81 5. Bordered determinants 81 6-7. The reduction of bilinear forms 82 8. Invariant factors 85 9. Vector products 86 10. The direct product 88 11. Induced or power matrices 89 12-14. Associated matrices 90 15. Transformable systems 93 16-17. Transformable linear sets 94 18-19. Irreducible transformable sets 99 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chVI.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices 102 I. Hermitian matrices 102 2. The invariant vectors of a hermitian matrix 104 3. Unitary and orthogonal matrices 105 4. Hermitian and quasi-hermitian forms 106 5. Reduction of a quasi-hermitian form 107 6. The Kronecker method of reduction 110 7. Cogredient transformation 112 8. Real representation of a hermitian matrix 114 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chVII.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices 116 I. Commutative matrices 116 2. Commutative sets of matrices 119 3. Rational methods 120 4. The direct product 122 5. Functions of commutative matrices 124 6. Sylvester's identities 125 7. Similar matrices 127 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chVIII.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices 129 I. Matric polynomials 129 2. Infinite series 129 3. The canonical form of a function 130 4. Roots of 0 and 1 132 5-6. The equation y^m = x; algebraic functions 133 7. The exponential and logarithmic functions 136 8. The canonical form of a matrix in a given field 137 9. The absolute value of a matrix 139 10. Infinite products 141 11. The absolute value of a tensor 141 12. Matric functions of a scalar variable 142 13. Functions of a variable vector 144 14. Functions of a variable matrix 149 15-16. Differentiation formulae 150 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter -1 coll17-chX.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras 154 I. Fields and algebras 154 2. Algebras which have a finite basis 155 3. The matric representation of an algebra 156 4. The calculus of complexes 157 5. The direct sum and product 158 6. Invariant subalgebras 159 7. Idempotent elements 161 8-9. Matric subalgebras 163 10-12. The classification of algebras 165 13. Semi-invariant subalgebras 170 14. The representation of a semi-simple algebra 172 15. Group algebras 174 Endmatter -1 coll17-bck.pdf 1 Frontmatter -1 Chapter I. Matrices and Vectors -1 Chapter II. Algebraic Operations with Matrices. The Charactersitic Equation -1 Chapter III. Invariant Factors and Elementary Divisors -1 Chapter IV. Vector Polynomials. Singular Matric Polynomials -1 Chapter V. Compound Matrices -1 Chapter VI. Symmetric, Skew, and Hermitian Matrices -1 Chapter VII. Commutative Matrices -1 Chapter VIII. Functions of Matrices -1 Chapter IX. The Automorphic Transformation of a Bilinear Form -1 Chapter X. Linear Associative Algebras -1 Endmatter 176 Appendix I 176 Notes 176 Appendix II 179 Bibliography 179 Index to Bibliography 201 Index 204