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A symmetric positive semi-definite matrix A is called completely positive ifthere exists a matrix B with nonnegative entries such that A=BB^T. If B is sucha matrix with a minimal number p of columns, then p is called the cp-rank of A.In this paper we develop a finite and exact algorithm to factorize any matrix Aof cp-rank 3. Failure of this algorithm implies that A does not have cp-rank 3.Our motivation stems from the question if there exist three nonnegativepolynomials of degree at most four that vanish at the boundary of an intervaland are orthonormal with respect to a certain inner product.

Factorization of CP-rank-3 completely positive matrices