Arithmetical properties of combinatorial matrices
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Results are derived on rational solutions to [...] where B is integral and A need not be square. It is shown that in general, provided a rational solution exists, one can be found in...
| Summary: | NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
Results are derived on rational solutions to [...] where B is integral and A need not be square. It is shown that in general, provided a rational solution exists, one can be found in which all denominators are a power of two. More general restrictions follow from the corresponding restrictions possible on rational lattices representing integral positive definite quadratic forms of determinant one. Results due to Kneser and others are applied to show that A may be taken as integral if it has no more than seven columns, half-integral if it has no more than sixteen columns.
These results are then applied to three types of matrix completion problems, integral matrices satisfying [...], partial Hadamard matrices and partial incidence matrices of symmetric block designs. It is found that rational normal completing matrices in which all denominators are powers of two are always possible in the first two cases and almost always possible in the final case.
Using a computer approach, the specific problem of showing that the last seven rows of a partial Hadamard matrix or a partial incidence matrix (with suitable parameters) can always be completed is tackled and it is shown that this is in fact the case, extending results by Marshall Hall for no more than four rows. An appendix lists the computer tabulation which is the basis of this conclusion.
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