Ivan Shishkin, Rye (1878)

Problems/TopologyUnreviewed

Path components of a set of matrices

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50
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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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  1. Show that Am=InA^m=I_n if and only if AA is diagonalizableFR with eigenvalues among the roots of unityFR, more precisely, among the mm th roots of unity.
  2. Show that GLn(C)\mathrm{GL}_n(\mathbb{C}) is path-connectedFR. Deduce that every similarity class is pathconnected.
  3. Count the path components of {A:Am=In}\left\{A: A^m=I_n\right\}. Work out the case m=2m=2.
  4. What does the result become for {A:∃m∈N∗,Am=In}\left\{A: \exists m \in \mathbb{N}^*, A^m=I_n\right\} ?
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