Completely Positive Integral Matrix And Its Application

The problem about completely positive matrices has been researched since 1961. Completely positive matrices have shown their importance in the study of block designs in combinational analysis, probability, inequality theory, statistics, economic models etc. Recall that an n x n matrix A is said to be completely positive if there exists a nonnegative n x m matrix B such that A=BBT. The smallest such number m is called the factorization index of A and denoted by cprankA. If B is a (0,1)-matrix, then A is called {0,1} completely positive and is denoted by {0.1}-cp. A is called r-uniform if B is a (0,1)-matrix with r 1s in each column, and called minimal if, for every nonzero nonnegative nxn diagonal matrix D, A-D is not {0,1}-cp. Similarly the factorization indices of nonnegative integral and that of {0,1}-cp matrices are respectively denoted by cprankz+A and cprank{o,1}A, in short, rankz+A and rank{0,1}A .To determine the existence of a completely positive integral factorization (or {0,1}- completely positive factorization) of a given matrix is a NP-Hard problem and has remained open till now. Moreover, to calculate the factorization index of a matrix has become even more challenging.This paper consists of four chapters. In chapter one, we introduce some notions and applications about integral completely positive matrices. Specifically we show their important application in block designs in detail. In chapter two, we discuss some characters of integral completely positive matrices by their definitions. In chapter three, we attempt starting with matrices with order no more than 4. We first characterize the integral factorizations and the indices (or {0,1}- ,completely positive factorizations) of matrices with order no more than 4. Then we prove that when this order n is 2 or 3, there exists a sufficient and necessary condition for a given nonnegative matrix with order n to be completely positive, and when n= 4, we turn to discuss several special matrices of this order. In chapter four, we first characterize r-uniform {0,1]-cp matrices, then we obtain some necessary conditions and sufficient conditions for a matrix to be minimal {0,1 }-cp. At last we present some bounds for {0,1 }-ranks.