Voltooiingsprobleme vir klasse van reële simmetriese matrikse wat geslote konvekse keëls vorm
The goal of any completion problem in matrix theory is to determine when a partial matrix can be completed to a matrix that conforms to certain conditions, where a partial matrix is a matrix with some unspecified entries. We consider the completion problem of a few classes of symmetric matrices which form closed convex cones. The completion problem of the SPN matrix, which is the sum of a positive semidefinite matrix and a nonnegative matrix, forms the main focus of this study. The completion problems of positive semidefinite matrices, completely positive matrices and SPN matrices are directly related to a certain graph that represents the partial matrices, namely the specification graph. It is shown that partial positive semidefinite matrices (and partial positive definite matrices) are completable if and only if the specification graph is chordal. In the case of a partial copositive matrix it is proved that all such matrices are completable to a copositive matrix. A greatest lower bound that depends on the diagonal entries is found for every unspecified entry. Since each completely positive matrix is positive semidefinite, stricter conditions than in the positive semidefinite case regarding the completion are required. A partial completely positive matrix is completable if and only if the specification graph is a block-clique graph, that is to say, each block in the graph is complete. For partial doubly nonnegative matrices it is seen that completions are possible under the same conditions as for the completely positive matrices. Finally, two equivalences for a matrix to be SPN completable are proved. The first one states that each cycle with odd length in the specification graph induces a complete subgraph. The second equivalence is in terms of the blocks of the specification graph: each block is either complete, bipartite or a Tk graph. completely positive completion, matrix completion problem, speci cation graph, chordal graph, block-clique graph. ii
SPN completion, Positive semidefinite completion, Copositive completion, Completely positive completion, Matrix completion problem, Specification graph, Chordal graph, Block-clique graph, SPN-voltooiing, Positiefsemidefiniete voltooiing, Kopositiewe voltooiing, Volledig-positiewe voltooiing, Matriksvoltooiingsprobleem, Spesifikasiegrafiek, Koordale grafiek, Blok-kliekgrafiek
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North-West University (South Africa), Potchefstroom Campus