Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The matricial relaxation of a linear matrix inequality   Given linear matrix inequalities (LMIs) L_1 and L_2, it is natural to ask:
(Q1) when does one dominate the other, that is, does L_1(X) PsD imply L_2(X)
PsD? (Q2) when do they have the same solution set? Such questions can be
NP-hard. This paper describes a natural relaxation of an LMI, based on
substituting matrices for the variables x_j. With this relaxation, the
domination questions (Q1) and (Q2) have elegant answers, indeed reduce to
constructible semidefinite programs. Assume there is an X such that L_1(X) and
L_2(X) are both PD, and suppose the positivity domain of L_1 is bounded. For
our "matrix variable" relaxation a positive answer to (Q1) is equivalent to the
existence of matrices V_j such that L_2(x)=V_1^* L_1(x) V_1 + ... + V_k^*
L_1(x) V_k. As for (Q2) we show that, up to redundancy, L_1 and L_2 are
unitarily equivalent.
  Such algebraic certificates are typically called Positivstellensaetze and the
above are examples of such for linear polynomials. The paper goes on to derive
a cleaner and more powerful Putinar-type Positivstellensatz for polynomials
positive on a bounded set of the form {X | L(X) PsD}.
  An observation at the core of the paper is that the relaxed LMI domination
problem is equivalent to a classical problem. Namely, the problem of
determining if a linear map from a subspace of matrices to a matrix algebra is
"completely positive".

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The categories of this paper are: solv-int
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The categories of this paper are: alg-geom
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The categories of this paper are: ao-sci
34
The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc