Cours 9 agreg interne réduction diagonalisation 3

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On tente ici de définir des fonctions inverses sur les matrices, comme la racine k-ième, le logarithme... On se rend compte que même si ces fonctions existent bien sur C, elles ne sont pas forcément définissables sur M_n(C). La diagonalisabilité nous permet de le faire à l'aide de polynômes interpolateurs de Lagrange. Dans le cas non diagonalisable, on s'en sort sur un ouvert, en utilisant la décomposition de Dunford et en remplaçant astuceieusement les séries de Taylor par des polynômes en une matrice 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