Journal
LINEAR ALGEBRA AND ITS APPLICATIONS
Volume 459, Issue -, Pages 465-474Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.laa.2014.07.017
Keywords
Vector configuration; Cone of positive semidefinite matrices; Von Neumann algebra; Trace
Categories
Funding
- FNS [200021-122084/1]
- OTKA [K 61116, NK 72523]
- MTA Renyi Lendulet Groups
- Graphs Research Group
- ERC Advanced Grant [227458]
- OTKA grant [104206]
- Bolyai Janos Research Scholarship of the Hungarian Academy of Sciences
- Swiss National Science Foundation (SNF) [200021_122084] Funding Source: Swiss National Science Foundation (SNF)
Ask authors/readers for more resources
Let v(1), ... ,v(n) be n vectors in an inner product space. Can we find a dimension d and positive (semidefinite) matrices A(1), ... , A(n) is an element of M-d(C) such that Tr(A(k)A(l)) = < v(k), v(l)> for all k, l = 1, ... , n? For such matrices to exist, one must have < v(k), v(l)> >= 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)>)((k,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. (C) 2014 Elsevier Inc. All rights reserved.
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