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How can I generate random commuting hermitian matrices ?

EDIT: Another question: given a certain hermitian matrix, how can I generate a random hermitian matrix which commutes with it?

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    $\begingroup$ The answer to the second question, as hinted out by the answer by drakas is to diagonalize the given matrix and use $U$ in his answer the unitary matrix composed of the eigenvectors. $\endgroup$
    – Tarek
    Commented Dec 4, 2012 at 14:21

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Take $m$ real diagonal matrices $D_m$ with $d_{kk}=\delta_{mk}$ and a unitary matrix $U\in \operatorname{U}(n)$. You'll get a set of $m$ commuting hermitian matrices by: $$ H_m=U^\dagger \cdot D_m \cdot U. $$
This is the maximal abelian subalgebra of the Lie algebra $\mathfrak{ u}(n)$. The centralizer of a maximal toral Lie subalgebra is called the Cartan subalgebra:

A Cartan subalgebra of the Lie algebra of $n×n$ matrices over a field is the algebra of all diagonal matrices.

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  • $\begingroup$ How can I generate a random unitary matrix ? $\endgroup$
    – Tarek
    Commented Dec 3, 2012 at 20:46
  • $\begingroup$ Use $\displaystyle U=e^{iH}$, where $H$ is a random hermitian matrix. $\endgroup$
    – draks ...
    Commented Dec 3, 2012 at 20:49
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    $\begingroup$ @draks... : could you please add a few words to explain the logical connections? What is $m$? (probably the number of dimensions, but then, it appears again as a subscript). In the sentence "this is the maximal", what is "this"? Of course, not $H_m$, as it should be grammatically. Why does the "centralizer" come into play? Is it relevant that one of the many Cartan subalgebra is made of the diagonal matrices? And so on. Thanks in advance. $\endgroup$ Commented Jan 13, 2023 at 20:20
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    $\begingroup$ Thanks, I perfectly understood this. What I would like to understand is the explanation in terms of maximal abelian subalgebra, centralizer, and Cartan subalgebra. $\endgroup$ Commented Jan 14, 2023 at 22:45

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