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Titre: Analyse matricielle algèbre linéaire appliquée



Résumé: Ce chapitre contient peu de démonstrations, son rôle est de fixer les notations et de rappeler les structures algébriques fondamentales, ainsi que les principaux résultats algébriques que nous utiliserons dans ce cours. Nous renvoyons le lecteur au cours de première année pour tout approfondissement.

Extrait du sommaire:

0. Préliminaires algébriques 1
1. Ensembles et applications 1
2. Les corps 2
3. Les anneaux 5
4. Les polynômes à une indéterminée 9
5. Arithmétique des polynômes 12
6. Les fractions rationnelles 19
I. Les matrices et abrégé d’algèbre linéaire 23
1. Les espaces vectoriels 1
1. La structure d’espace vectoriel 1
2. Bases et dimension d’un espace vectoriel 5
3. Somme de sous-espaces vectoriels 7
4. Les applications linéaires 9
5. Exercices 15
2. Les matrices 1
1. Définitions 1
2. Produit de matrices 5
3. Matrice d’une application linéaire 10
4. Trace d’une matrice 15
5. Noyau et image d’une matrice 15
6. Le rang d’une matrice 17
7. Opérations matricielles par blocs 18
8. Exercices 21
3. Les déterminants 1
1. Définition récursive du déterminant 1
2. Premières propriétés du déterminant 3
3. Les formules de Cramer 8
4. Formulation explicite du déterminant 10
5. Calcul des déterminants 12
6. Calcul de l’inverse d’une matrice 15
7. Déterminant d’un endomorphisme 17
8. Annexe : rappels sur les groupes de symétries 18
9. Annexe : déterminants et formes multilinéaires alternées 20
II. La réduction des matrices 23
4. Pour se mettre en appétit 1
1. Équations d’évolution linéaire couplées 1
2. Le découplage de système d’équations 5
3. La diagonalisation des matrices et des endomorphismes 8
4. Marches sur un graphe et diagonalisation 11
5. Exercices 14
5. Valeurs propres et vecteurs propres 1
1. Préliminaires 1
2. Valeurs propres et espaces propres 5
3. Calcul des valeurs propres 9
4. Le cas des endomorphismes 11
5. Exercices 13
6. Trigonalisation et diagonalisation 1
1. Trigonalisation des matrices 1
2. Diagonalisation des matrices 9
3. Une obstruction au caractère diagonalisable 12
4. Caractérisation des matrices diagonalisables 15
5. Matrices diagonalisables : premières applications 17
6. Trigonalisation et diagonalisation des endomorphismes 20
7. Exercices 24
7. Le polynôme minimal 1
1. Préliminaires 1
2. Polynômes de matrices 3
3. Le lemme de décomposition en noyaux 6
4. Le polynôme minimal 11
5. Le théorème de Cayley-Hamilton 14
6. Le cas des endomorphismes 21
7. Exercices 24
8. Décomposition spectrale des matrices 1
1. Préliminaires 1
2. Matrices nilpotentes 3
3. Les espaces spectraux 4
4. Décomposition spectrale géométrique 7
Table des matières 1
5. Décomposition spectrale algébrique 10
6. Calcul de la décomposition spectrale algébrique 15
7. Exercices 18
III. Applications de la réduction des matrices 23
9. Fonctions de matrices 1
1. Calcul des puissances d’une matrice 1
2. La fonction exponentielle 4
3. Exercices 7
10. Systèmes dynamiques discrets 1
1. Les suites récurrentes 1
2. La suite de Fibonacci (1202) 3
3. Dynamique de populations 4
4. Exercices 7
11. Systèmes dynamiques continus 1
1. Systèmes différentiels linéaires à coefficients constants 2
2. Exemples 7
3. Exercices 14
Bibliographie 1

Mathématique appliquée cours 1

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