| 词条 | Draft:Matrix completions |
| 释义 |
Another natural question for the completion theory of a given property Q is for which patterns of the specified entries does every partial Q matrix have a completion with property Q? For this purpose it is natural to describe the location of specified entries in an n-by-n partial matrix A = (ai,j) with a graph G(A) in which edges correspond to specified entries. For example, if the property naturally involves symmetry (as when Q is positive definiteness), G will be an undirected graph on n vertices in which i, j is an edge if and only if ai,j is specified. Usually, the diagonal entries of A are naturally assumed to be specified but loops in G are suppressed. Whenever there is assumed symmetry in the placement of data (even if the property does not imply symmetry), an undirected graph might be used, but in other square problems a directed graph could be used; in nonsquare problems a bipartite graph (rows vs. columns) is after natural. When Q is positive definiteness, for example, the graphs for which all partial positive definite matrices have positive definite completions are exactly the chordal graphs. Minimal rank completions have become very useful in predicting consumer preferences, and was an underlying tool in the successful submission of the Netflix Prize that was awarded on September 21, 2009. References
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