Matrix scaling, entropy minimization, and conjugate duality. I: Existence conditions
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.
scientific article; zbMATH DE number 4101373
scientific article