Ivan Shishkin, Rye (1878)

Problems/Linear algebraExerciseUnreviewed

Step by step proof of the spectral theorem

by Anduril·
35
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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.

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Let A∈Sn(R)A \in S_{n}(\mathbb{R}) and uu an endomorphism of Rn\mathbb{R}^{n} so that MatB(u)=AMat_{B}(u)=A with BB the canonic basis of Rn\mathbb{R}^{n}.

  1. Show that a real endomorphism stabilizes at least space of dimension 1 or 2.
  2. Show that Rn\mathbb{R}^{n} can be written as a sum of orthogonal spaces of dimension 1 or 2 stabilized by uu.
  3. Show the spectral theorem in dimension 2.
  4. Prove the Spectral Theorem.
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