Greatest common divisor From Wikipedia, the free encyclopedia Contents 1 Augmented matrix 1 1.1 Examples ............................................... 1 1.1.1 Matrix inverse ......................................... 1 1.1.2 Existence and number of solutions .............................. 2 1.1.3 Solution of a linear system .................................. 2 1.2 References ............................................... 3 2 Coefficient 4 2.1 Linear algebra ............................................. 4 2.2 Examples of physical coefficients ................................... 5 2.3 See also ................................................ 5 2.4 References ............................................... 5 3 Coefficient matrix 6 3.1 Example ................................................ 6 3.2 See also ................................................ 6 4 Determinant 7 4.1 Definition ............................................... 7 4.1.1 2 × 2 matrices ......................................... 8 4.1.2 3 × 3 matrices ......................................... 10 4.1.3 n × n matrices ......................................... 10 4.2 Properties of the determinant ..................................... 12 4.2.1 Multiplicativity and matrix groups .............................. 14 4.2.2 Laplace’s formula and the adjugate matrix .......................... 14 4.2.3 Sylvester’s determinant theorem ............................... 15 4.3 Properties of the determinant in relation to other notions ....................... 15 4.3.1 Relation to eigenvalues and trace ............................... 15 4.3.2 Cramer’s rule ......................................... 17 4.3.3 Block matrices ........................................ 17 4.3.4 Derivative ........................................... 18 4.4 Abstract algebraic aspects ....................................... 19 4.4.1 Determinant of an endomorphism .............................. 19 i ii CONTENTS 4.4.2 Exterior algebra ........................................ 19 4.4.3 Square matrices over commutative rings and abstract properties ............... 20 4.5 Generalizations and related notions .................................. 21 4.5.1 Infinite matrices ........................................ 21 4.5.2 Related notions for non-commutative rings .......................... 21 4.5.3 Further variants ........................................ 21 4.6 Calculation ............................................... 21 4.6.1 Decomposition methods ................................... 21 4.6.2 Further methods ....................................... 22 4.7 History ................................................. 22 4.8 Applications .............................................. 23 4.8.1 Linear independence ..................................... 23 4.8.2 Orientation of a basis ..................................... 23 4.8.3 Volume and Jacobian determinant .............................. 24 4.8.4 Vandermonde determinant (alternant) ............................ 24 4.8.5 Circulants ........................................... 25 4.9 See also ................................................ 25 4.10 Notes ................................................. 25 4.11 References ............................................... 27 4.12 External links ............................................. 27 5 Greatest common divisor 28 5.1 Overview ............................................... 28 5.1.1 Notation ........................................... 28 5.1.2 Example ........................................... 28 5.1.3 Reducing fractions ...................................... 29 5.1.4 Coprime numbers ...................................... 29 5.1.5 A geometric view ....................................... 29 5.2 Calculation .............................................. 29 5.2.1 Using prime factorizations .................................. 29 5.2.2 Using Euclid’s algorithm ................................... 30 5.2.3 Binary method ........................................ 30 5.2.4 Other methods ........................................ 31 5.3 Properties ............................................... 32 5.4 Probabilities and expected value ................................... 33 5.5 The gcd in commutative rings ..................................... 33 5.6 See also ................................................ 34 5.7 Notes ................................................. 34 5.8 References .............................................. 35 5.9 Further reading ............................................ 35 5.10 External links ............................................. 35 CONTENTS iii 6 Linear equation 38 6.1 One variable ............................................. 39 6.2 Two variables ............................................. 39 6.2.1 Forms for two-dimensional linear equations ......................... 39 6.2.2 Connection with linear functions ............................... 43 6.2.3 Examples ........................................... 44 6.3 More than two variables ........................................ 44 6.4 See also ................................................ 44 6.5 Notes ................................................. 44 6.6 References .............................................. 44 6.7 External links ............................................. 45 7 Mathematics 46 7.1 History ................................................. 47 7.1.1 Evolution ........................................... 47 7.1.2 Etymology .......................................... 49 7.2 Definitions of mathematics ...................................... 50 7.2.1 Mathematics as science .................................... 50 7.3 Inspiration, pure and applied mathematics, and aesthetics ....................... 53 7.4 Notation, language, and rigor ..................................... 54 7.5 Fields of mathematics ......................................... 54 7.5.1 Foundations and philosophy .................................. 55 7.5.2 Pure mathematics ....................................... 56 7.5.3 Applied mathematics ..................................... 58 7.6 Mathematical awards ......................................... 58 7.7 See also ................................................ 59 7.8 Notes ................................................. 59 7.9 References ............................................... 61 7.10 Further reading ............................................ 62 7.11 External links ............................................. 62 8 Matrix (mathematics) 64 8.1 Definition ............................................... 65 8.1.1 Size .............................................. 65 8.2 Notation ................................................ 66 8.3 Basic operations ............................................ 66 8.3.1 Addition, scalar multiplication and transposition ....................... 66 8.3.2 Matrix multiplication ..................................... 67 8.3.3 Row operations ........................................ 68 8.3.4 Submatrix ........................................... 68 8.4 Linear equations ............................................ 69 8.5 Linear transformations ......................................... 69 iv CONTENTS 8.6 Square matrices ............................................ 69 8.6.1 Main types .......................................... 70 8.6.2 Main operations ........................................ 72 8.7 Computational aspects ......................................... 73 8.8 Decomposition ............................................ 74 8.9 Abstract algebraic aspects and generalizations ............................. 75 8.9.1 Matrices with more general entries .............................. 75 8.9.2 Relationship to linear maps .................................. 76 8.9.3 Matrix groups ......................................... 76 8.9.4 Infinite matrices ........................................ 77 8.9.5 Empty matrices ........................................ 77 8.10 Applications .............................................. 78 8.10.1 Graph theory ......................................... 78 8.10.2 Analysis and geometry .................................... 78 8.10.3 Probability theory and statistics ................................ 79 8.10.4 Symmetries and transformations in physics .......................... 80 8.10.5 Linear combinations of quantum states ............................ 81 8.10.6 Normal modes ........................................ 81 8.10.7 Geometrical optics ...................................... 82 8.10.8 Electronics .......................................... 82 8.11 History ................................................. 82 8.11.1 Other historical usages of the word “matrix” in mathematics ................. 83 8.12 See also ................................................ 83 8.13 Notes ................................................. 84 8.14 References ............................................... 87 8.14.1 Physics references ....................................... 89 8.14.2 Historical references ..................................... 90 8.15 External links ............................................. 90 9 Numerical linear algebra 92 9.1 See also ................................................ 92 9.2 References .............................................. 92 9.3 External links ............................................. 93 10 System of linear equations 94 10.1 Elementary example .......................................... 95 10.2 General form ............................................