Matrix scaling, entropy minimization, and conjugate duality. I: Existence conditions
type: Item
instance of: scholarly article
title: Matrix scaling, entropy minimization, and conjugate duality. I: Existence conditions
zbMATH Open document ID: 0673.15003
DOI: 10.1016/0024-3795(89)90493-X
author: Michael H. Schneider
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: An \(n\times n\) matrix A is said to be balanced if the ith row sum equals the ith column sum, \(i=1,...,n\). The author considers three problems. (1) Given a nonnegative \(n\times n\) matrix A, find a positive definite diagonal matrix D so that \(DAD^{-1}\) is balanced. (2) Given a nonnegative \(n\times n\) matrix A and positive vectors of length m and n, respectively, find positive definite diagonal matrices \(D_ 1\) and \(D_ 2\) of appropriate dimension so that \(D_ 1AD_ 2\) has row sums \(\alpha_ 1\) and column sums \(\alpha_ 2\). (3) Given nonnegative \(n\times n\) matrices A, L, and U where \(L\leq U\) elementwise, find a nonnegative matrix \(\Lambda\) and a positive definite diagonal matrix D, such that the Hadamard product X of \(\Lambda\) and \(DAD^{-1}\) satisfies \(x_{ij}=\ell_{ij}\) whenever \(\lambda_{ij}>1\), \(x_{ij}=u_{ij}\) whenever \(\lambda_{ij}<1\) and in general \(L\leq X\leq U\) elementwise. Some results concerning existence of solutions to these problems are developed. A very extensive bibliography is cited concerning other researches on these problems.
Mathematics Subject Classification ID: 15A12
zbMATH DE Number: 4101373
zbMATH Keywords: matrix scaling
zbMATH Keywords: entropy minimization
zbMATH Keywords: conjugate duality
zbMATH Keywords: balanced
zbMATH Keywords: matrix
zbMATH Keywords: Hadamard product
zbMATH Keywords: bibliography
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
cites work: Q5592025
cites work: On the RAS-algorithm
cites work: Algorithms for proportional matrices in reals and integers
cites work: An Axiomatic Approach to Proportionality Between Matrices
cites work: An extension of a theorem of Darroch and Ratcliff in loglinear models and its application to scaling multidimensional matrices
cites work: Convex Sets of Non-Negative Matrices
cites work: Line-sum-symmetric scalings of square nonnegative matrices
cites work: On some methods for entropy maximization and matrix scaling
cites work: A note on solution of large sparse maximum entropy problems with linear equality constraints
cites work: Entropy in linear programs
cites work: Optimal spatial interaction and the gravity model
cites work: On the scaling of multidimensional matrices
cites work: Matrix balancing
cites work: Q3274170
cites work: Properties of Kruithof's Projection Method
cites work: Q3292914
cites work: Scaling of matrices to achieve specified row and column sums
cites work: Matrix Links, An Extremization Problem, and the Reduction of a Non-Negative Matrix to One With Prescribed Row and Column Sums
cites work: The spectrum of a nonlinear operator associated with a matrix
cites work: On Pre-Conditioning of Matrices
cites work: Q5185585
cites work: Convex Analysis
cites work: Generalized scalings satisfying linear equations
cites work: Scalings of matrices which have prespecified row sums and column sums via optimization
cites work: Matrix scaling, entropy minimization, and conjugate duality. II: The dual problem
cites work: A Comparative Study of Algorithms for Matrix Balancing
cites work: A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices
description: scientific article; zbMATH DE number 4101373