Scaling Matrices to Prescribed Row and Column Maxima

A nonnegative symmetric matrix $B$ has row maxima prescribed by a given vector $r$, if for each index $i$, the maximum entry in the $i$th row of $B$ equals $r_i$. This paper presents necessary and sufficient conditions so that for a given nonnegative symmetric matrix $A$ and positive vector $r$ there exists a positive diagonal matrix $D$ such that $B = DAD$ has row maxima prescribed by $r$. Further, an algorithm is described that either finds such a matrix $D$ or shows that no such matrix exists. The algorithm requires $O(n \lg n + p)$ comparisons, $O(p)$ multiplications and divisions, and $O(q)$ square root calculations where $n$ is the order of the matrix, $p$ is the number of its nonzero elements, and $q$ is the number of its nonzero diagonal elements. The solvability conditions are compared and contrasted with known solvability conditions for the analogous problem with respect to row sums. The results are applied to solve the problem of determining for a given nonnegative rectangular matrix $A$ positive, diagonal matrices $D$ and $E$ such that $DAE$ has prescribed row and column maxima. The paper presents an equivalent graph formulation of the problem. The results are compared to analogous results for scaling a nonnegative matrix to have prescribed row and column sums and are extended to the problem of determining a matrix whose rows have prescribed $l_p$ norms.