Ranks of Completions of Partial Matrices

For an n-by-m array with some entries specified and the remainder free to be chosen from a given field, we study the possible ranks occurring among all completions. For any such partial matrix the maximum rank may be nicely characterized and all possible ranks between the minimum and maximum are attained. The minimum is more delicate and is not in general determined just by the ranks of fully specified submatrices. This focusses attention upon the patterns of specified entries for which the minimum is so determined. It is shown that it is necessary that the graph of the pattern be (bipartite) chordal, and some evidence is given for the conjecture that this is also sufficient.

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