论文标题
关于完全阳性矩阵的CP因数的数量
On the number of CP factorizations of a completely positive matrix
论文作者
论文摘要
如果$ a = bb^t $，则方形矩阵$ a $是完全积极的，其中$ b $是（不一定是方形）非负矩阵。通常，完全阳性的矩阵可能具有许多这样的CP因素。但是在某些情况下，存在独特的CP分解。我们证明了一个完全正面矩阵的简单必要条件，该矩阵的三角形是免费的，可以进行独特的CP分解。这意味着在圆锥$ \ Mathcal {cp} _n $ of $ n \ times n $完全正矩阵的边界上的其他矩阵上的CP分解的唯一性。我们还描述了$ \ Mathcal {cp} _n $包含完全正$ a $的最小面孔。如果$ a $具有独特的CP分解，则该面部是多面体。
A square matrix $A$ is completely positive if $A=BB^T$, where $B$ is a (not necessarily square) nonnegative matrix. In general, a completely positive matrix may have many, even infinitely many, such CP factorizations. But in some cases a unique CP factorization exists. We prove a simple necessary and sufficient condition for a completely positive matrix whose graph is triangle free to have a unique CP factorization. This implies uniqueness of the CP factorization for some other matrices on the boundary of the cone $\mathcal{CP}_n$ of $n\times n$ completely positive matrices. We also describe the minimal face of $\mathcal{CP}_n$ containing a completely positive $A$. If $A$ has a unique CP factorization, this face is polyhedral.
