Diagonal equivalence to matrices with prescribed row and column sums. II

Abstract

Let A A be a nonnegative m × n m \times n matrix and let r = ( r 1 , ⋯ , r m ) r = ({r_1}, \cdots ,{r_m}) and c = ( c 1 , ⋯ , c n ) c = ({c_1}, \cdots ,{c_n}) be positive vectors such that Σ i = 1 m r i = Σ j = 1 n c j \Sigma _{i = 1}^m{r_i} = \Sigma _{j = 1}^n{c_j} . It is well known that if there exists a nonnegative m × n m \times n matrix B B with the same zero pattern as A A having the i i th row sum r i {r_i} and j j th column sum c j {c_j} , there exist diagonal matrices D 1 {D_1} and D 2 {D_2} with positive main diagonals such that D 1 A D 2 {D_1}A{D_2} has i i th row sum r i {r_i} and j j th column sum c j {c_j} . However the known proofs are at best cumbersome. It is shown here that this result can be obtained by considering the minimum of a certain real-valued function of n n positive variables.