On classes of normalized matrices

Abstract

The integer m × n matrices A = ( a ij ), B = ( b ij ) are said to be equivalent if a ij = u i + b ij + v j for all i = 1,…, m, j = 1,…, n, for some u 1,…, u m , v 1,…, v n ∈ Z. For an integer matrix X the symbol S( X) denotes the set of all nonnegative integer matrices equivalent to X having a zero element in each row and each column. We develop algorithms to solve the problems of the following type for a given class S( X) : decide whether A ∈ S( X); find the largest possible value for each position among all matrices in S( X); find a matrix in S( X) with prescribed values of a specified entry or row (column); find a matrix in S( X) with term rank 2. A complete description of those S( X) containing only zero-one matrices is presented.