The real positive definite completion problem cycle completability.pdf

THE REAL POSITIVE DEFINITE COMPLETION PROBLEMFOR A SIMPLE CYCLE*WAYNE BARRETT**, CHARLES R. JOHNSONy and PABLO TARAZAGAzAbstract. We consider the question of whether a real partial positive denite matrix (in which thespecied o-diagonal entries consist of a full n cycle) has a positive denite completion. This lies incontrast to the previously studied chordal case. We give two solutions. In one, we describe about n2independent conditions on angles associated with a normalization of the data that are necessary andsucient. The second is more computational and allows presentation of all positive denite completions,as well as answering the existence question.1. Introduction. A real partial matrix A is one in which some entries are speciedreal numbers and the remainder are unspecied, i.e., free variables over the real numbers.We say that A is partial symmetric if A is square, aji is specied whenever aij is, andaji = aij . We shall assume throughout that the diagonal entries of A are specied. Anexample is(1) A = 24 5 2 ?2 1 2? 2 335in which the ?s indicate unspecied entries.A partial positive denite matrix is a partial symmetric matrix each of whose speciedprincipal submatrices is positive denite. (By a specied portion of a partial matrix wealways mean one composed entirely of specied entries.) Partial positive semidenitematrices are dened similarly. The matrix above is not partial positive denite, but isif the 2,2 entry is replaced, for example, by 2. A completion of a partial matrix is aspecication of the unspecied entries resulting in a conventional matrix, and the positivedenite completion problem is to determine if a positive denite completion exists or tond a completion of a partial positive denite matrix that is positive denite. For example,24 5 2 12 2 21 2 335 is a positive denite completion of 24 5 2 ?2 2 2? 2 335.*This manuscript was prepared while the rst two authors were visitors at the Institute for Mathematicsand its Applications, M