Positive definite completions of partial Hermitian matrices |
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Authors: | Robert Grone Charles R Johnson Eduardo M Sá Henry Wolkowicz |
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Institution: | Department of Mathematics Auburn University, Alabama 36849, USA;Institute for Physical Science and Technology University of Maryland College Park, Maryland 20742, USA;Departmento de Matematica Universidade de Aveiro Aveiro, Portugal;Department of Mathematics University of Alberta Edmonton, Alberta, T6G 2G1, Canada |
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Abstract: | The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made. |
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