Bapat, R. B. ; Raghavan, T. E. S.
(1989)
*An extension of a theorem of Darroch and Ratcliff in loglinear models and its application to scaling multidimensional matrices*
Linear Algebra and its Applications, 114-115
.
pp. 705-715.
ISSN 0024-3795

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Official URL: http://www.sciencedirect.com/science/article/pii/0...

Related URL: http://dx.doi.org/10.1016/0024-3795(89)90489-8

## Abstract

Let C=(c_{ij}) be an m ×n matrix with real entries. Let b be any nonzero m-vector. Let K = {ΠCΠ = b, Π ≥ 0} be bounded. Let x = (x_{1}, x_{2},...,x_{n}), y = (y_{1}, y_{2},...,y_{n}) be two nonnegative vectors with y ∈ K and x_{j} = 0 ⇔ y_{j} = 0 for any coordinate j. Then it is shown that there exists a Π ∈ K and positive numbers z_{1}, z_{2},..., z_{m} such that π_{j} = xj∈mi = 1 zciji for all j. Th theorem slightly generalizes a theorem of Darroch and Ratcliff in loglinear models with a completely different proof technique. The proof relies on an extension of a topological theorem of Kronecker to set valued maps and the duality theorem of linear programming. Many theorems in scaling of matrices and multidimensional matrices are direct consequences of this theorem. The main idea is to associate a suitable zero-one matrix of transportation with any multidimensional matrix. Some motivations for scaling applications are also discussed.

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Deposited On: | 14 Jan 2012 15:24 |

Last Modified: | 14 Jan 2012 15:24 |

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