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Let $h$ be a positive definite hermitian form on $E$ and $A,B: E\rightarrow E$ be two hermitian endomorphisms which commute, $AB = BA$. Prove that there exists a orthogonal basis for $E$ consisting of common eigenvectors for $A$ and $B$.

Serge Lange,"Algebra" Chapter 15,Question 4

Can you please give any hint?

Hitendra Kumar
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  • https://math.stackexchange.com/questions/6258/matrices-commute-if-and-only-if-they-share-a-common-basis-of-eigenvectors – Federico Nov 05 '18 at 18:18
  • But there we are not using that $A,B$ are hermitian endomorphism. And we want a orthogonal basis. Do we get orthoganal basis using orthogonality process?? – Hitendra Kumar Nov 06 '18 at 04:58
  • hermitian implies that they are individually diagonalizable. then commutativity is equivalent to having a common basis of eigenvectors. the orhogonality follows from the fact that the eigenspaces are ortghogonal – Federico Nov 06 '18 at 10:56

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