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

Frenkel, Péter and Weiner, Mihály (2014) On vector configurations that can be realized in the cone of positive matrices. LINEAR ALGEBRA AND ITS APPLICATIONS, 459. pp. 465-474. ISSN 0024-3795

Preview

Text vectorconf.pdf Download (292kB) | Preview

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 A1,..., An of size d×d 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 < vk,vl > ≥ 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.

Actions (login required)

Edit Item