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On vector configurations that can be realized in the cone of positive matrices

Weiner, Mihály
Year 2010
Description 8 p.
Abstract Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a natural number $d$ and positive (semidefinite) complex matrices $A_1$,..., $A_n$ of size $d imes d$ such that ${ m Tr}(A_kA_l)= <v_k, v_l>$ for all $k,l=1,..., n$? For such matrices to exist, one must have $<v_k, v_l> geq 0$ for all $k,l=1,..., n$. We prove that if $n<5$ then this trivial necessary condition is also a sufficient one and find an appropriate example showing that from $n=5$ this is not so --- even if we allowed realizations by positive operators in a von Neumann algebra with a faithful normal tracial state. The fact that the first such example occurs at $n=5$ is similar to what one has in the well-investigated problem of positive factorization of positive (semidefinite) matrices. If the matrix $(<v_k, v_l>)$ has a positive factorization, then matrices $A_1$,..., $A_n$ as above exist. However, as we show by a large class of examples constructed with the help of the Clifford algebra, the converse implication is false.
Stable URL http://archive-ouverte.unige.ch/unige:11987
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arXiv: 1004.0686

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Deposited on : 2010-10-01

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