Aperçu des sections

  • Algebra 3 (Reduction of endomorphismsmes)

    This course is intended for students of the second year LMD Mathematics who have already taken their course in linear algebra of the first year. It consists of Algebra 3 which deals with the subject of the diagonalization of endomorphisms (of square matrices) and their applications to some mathematical aspects. The reduction of endomorphisms is a powerful tool for the determination of several notions of linear algebra such as the rank of a matrix, the power of a matrix, the inverse of an invertible matrix, etc. As it has several applications in other mathematical aspects, such as the study of inverse problems, linear recurrent sequences, the resolution of differential equations and the exponential of a matrix. The aim of this part is to expose the theorem of the diagonalization of endomorphisms and to present it to students of the second year LMD Mathematics in a simpler and more understandable course.

    • Table of Contents

      1 Some preliminaries for the diagonalization of endomorphisms  

      Invariant vector subspaces 16 Matrix representation  16 Cyclic subspaces 17Characteristic subspaces  18 The kernel lemma 19 Series of exercises .

      2 Vector subspaces invariant by an endomorphism 

       3 Eigenvalues ​​and Eigenvectors 23 Eigenvectors and stable subspaces  24 Characteristic polynomial Cayley-Hamilton Theorem 26 Minimal Polynomial  27Exercise Series  29

      4 Canonical forms  The rational form (Frobenius form) 33 The Jordan normal form  34 Similarity of a diagonal Jordan block matrix 35 Proof of the theorem  35The Jordan normal form. 37

      Series of exercises .

      5 Applications of the diagonalization of endomorphisms 

      1) Application to linear recurrent sequences 41

      2) Application to linear differential equations 43

      • The contents

      • corrigé exercice 48

      • Correction des exercices