In this thesis, we will investigate the completely positive matrices and their completely positive rank. The problems of determining whether a matrix is completely positive and computing its completely positive rank are still open. We will first present the main definitions and known results of this topic. We will also present M-matrices, diagonally dominant matrices, and discuss the geometric approach to complete positivity. Furthermore, we will take a look at the alternative procedure of finding constraints of the completely positive rank of matrices using graph theory. In particular, we will define the characteristics of the corresponding graphs which bound the completely positive rank of the matrix.
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