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Abstract
Necessary and sufficient conditions are proven for the existence of a square matrix, over an arbitrary field, such that for every principal submatrix the sum of the elements in the row complement of the submatrix is prescribed. The problem is solved in the cases where the positions of the nonzero elements of A are contained in a given set of positions, and where there is no restriction on the positions of the nonzero elements of A. The uniqueness of the solution is studied as well. The results are used to solve the cases where the matrix is required to be symmetric and/or nonnegative entrywise.
*The research of this author was carried out within the activity of the Centro de Algebra da Universidade de Lisboa
†The research of these authors was supported by their grant No. 90-00434 from the United States -Israel Binational Science Foundation, Jerusalem, Israel
‡The research of this author was supported in part by NSF grant DMS-9123318. The research of all three authors was supported by the Fundacao Calouste Gulbenkian and STRIDE project, Lisboa
*The research of this author was carried out within the activity of the Centro de Algebra da Universidade de Lisboa
†The research of these authors was supported by their grant No. 90-00434 from the United States -Israel Binational Science Foundation, Jerusalem, Israel
‡The research of this author was supported in part by NSF grant DMS-9123318. The research of all three authors was supported by the Fundacao Calouste Gulbenkian and STRIDE project, Lisboa
Notes
*The research of this author was carried out within the activity of the Centro de Algebra da Universidade de Lisboa
†The research of these authors was supported by their grant No. 90-00434 from the United States -Israel Binational Science Foundation, Jerusalem, Israel
‡The research of this author was supported in part by NSF grant DMS-9123318. The research of all three authors was supported by the Fundacao Calouste Gulbenkian and STRIDE project, Lisboa