This article characterizes the rank-one factorization of auto-correlation matrix polynomials. We establish a sufficient and necessary uniqueness condition for uniqueness of the factorization based on the greatest common divisor (GCD) of multiple polynomials. In the unique case, we show that the factorization can be carried out explicitly using GCDs. In the non-unique case, the number of non-trivially different factorizations is given and all solutions are enumerated.

翻译：本文刻画了自相关矩阵多项式的秩一分解。我们基于多个多项式的最大公约数（GCD），建立了分解唯一性的充要条件。在唯一的情况下，我们证明可以通过GCD显式地进行分解。在非唯一的情况下，给出了非平凡不同分解的个数，并枚举了所有解。