Can Commuting Operators in a Hilbert Space Share a Common Set of Eigenvectors?

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The discussion centers on the analytical methods for finding a common set of eigenvectors for commuting operators in a d-dimensional Hilbert space. Specifically, it addresses the challenge of computing an orthonormal basis composed of d eigenvectors that are shared among n commuting operators, represented as d x d complex square matrices. The inquiry suggests exploring computational tools like Mathematica for efficient calculations, with a reference to a potential resource from RWTH Aachen University.

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Given a set of n<d commuting operators, either degenerate or non-degenerate, in a d-dimensional Hilbert space, is there an effective analytical method of finding an orthonormal basis composed of d eigenvectors common to all the operators in the set?
The operators are dxd complex square matrices, and the d-dimensional vectors in the desired orthonormal basis must be eigenvectors of all the operators. I was wondering if there is an efficient way to compute such vectors in a computer algebra system, such as Mathematica.
 
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I don't know, but maybe http://www.math.rwth-aachen.de/mapleAnswers/html/368.html helps.
 
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