Completely Positive Matrices Of Order Five

An nxn matrix A is called doubly nonnegative if A is both entrywise nonnegative and positive semidefinite as well. A is called completely positive if A can be factorized as A = BB~T, where B is an nxm entrywise nonnegative matrix for some integer m. The smallest such number m is called the factorizationindex (or the cp-rank) of A. We use DNN_n to denote the set of all nxn doublynonnegative matrices and CP_n the set of all nxn completely positive matrices. The cp-rank of A is denoted by cprank(A). A graph G of order n is called cp if for each matrix A (A∈DNN_n with G(A)≡ G is cp. Recall that a DAW (doublynonnegative) realization of a graph G is defined to be a matrix A∈DNN_n whoseassociate graph is G ; Analogically, we define cp, nonnegative, and positive (semi) definite realizations of G .Early in 1963, Maxfield and Minc proved, by using matrix theory, that a matrix of order less than 5 is completely positive if and only if it is doubly nonnegative. They also gave a counterexample to show that this is not the casefor matrices of order greater than 4, i.e., CP_n (?) DNN_n for n ≥ 5 . Study oncomplete positivity of some special matrices began in 1987. In 1988, Kaykobad used the graphic method to prove that all symmetric nonnegative diagonally dominant matrices are cp. In 1991, Ando showed that a positive semidefinite Hankel matrix is cp if the submatrix obtained from A by deleted the first column and the last row is also positive semidefinite. In 1994, Drew et al. showed that a symmetric nonnegative matrix A is cp if its comparison matrix is an M-matrix. Characterization of cp graphs is fulfilled by A.Berman etc. They showed that the following four conditions are identical:1) A graph G is a cp graph.2) Each block of G is cp.3) Each block of G is isomorphic either to aK4 (the complete graph of order 4), or to a Tk (k triangles with a common base) or to a bipartite.4)G does not contain any long odd cycle (odd cycle of length greater than 4).5) G is a line perfect graph.From the department history of more than forty years of the research of completely positive matrices, we can see thai the question of completely positive matrices of order five is the key and watershed of the research of the question of completely positive matrices. Exactly because of this reason, we mainly investigate the question of completely positive matrices of doubly nonnegative matrices of order five. We wish get the hole of the question of common doubly nonnegative matrices. In this thesis, we focus on the problem of complete positivity of doubly nonnegative matrices of order five. This thesis has four sections. In Section 1 we introduce the background of completely positive matrices, several equivalent definitions for cp and relevant concepts. Then we restate the three basic problems concerning completely positive matrices.1) Finding a feasible sufficient and necessary condition to discuss a matrix being completely positive;2) Computing the factorization index of completely positive matrices;3) Given a graph G , computingBy far, no question has more complete solution.In Section 2 we introduce some development on cp problem and give some necessary and sufficient conditions for small matrices to be cp; Characterizations of cp graphs and that of several special kinds of completely positive matrices are also presented here. Section 3 and 4 are the main parts of the thesis: In Section 3, we first classify the matrices in DNN5 according to the associate non-cp graphs oforder five, and use elementary transformations on matrices to describe necessary and sufficient conditions of doubly nonnegative realizations of the first five classesto be cp, and obtain necessary and sufficient conditions for doubly nonnegative realizations of the 6th class to be cp, by employing some known results on book graphs. In Section 4, we deal with the last two classes of matrices, and transform the problem into the familiar cases we have coped with in Section 3, by using the techniques related to Schur complement, edge-deleling process, and geometry.