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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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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. –  Tarek Dec 4 '12 at 14:21
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1 Answer

up vote 1 down vote accepted

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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How can I generate a random unitary matrix ? –  Tarek Dec 3 '12 at 20:46
    
Use $\displaystyle U=e^{iH}$, where $H$ is a random hermitian matrix. –  draks ... Dec 3 '12 at 20:49
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