Ivan Shishkin, Rye (1878)

Problems/Linear algebra

Un polynôme de matrice

by Évariste d'aubergine·
30
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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ReviewedReviewed by Ancient Tree
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Soit M∈Sn(R)M \in \mathbb S_n(\mathbb R) telle que
M5+M4+M3+M2+M=0M^5 +M^4 + M^3+M^2 +M=0

  1. MM est elle diagonalisable?
  2. Soit λ\lambda une valeur propre de MM. Montrer que λ5+λ4+λ3+λ2+λ=0\lambda^5 + \lambda^4 +\lambda^3 +\lambda^2 + \lambda =0, puis en déduire que M=0M=0.
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