We investigate rank revealing factorizations of $m \times n$ polynomial matrices $P(\lambda)$ into products of three, $P(\lambda) = L(\lambda) E(\lambda) R(\lambda)$, or two, $P(\lambda) = L(\lambda) R(\lambda)$, polynomial matrices. Among all possible factorizations of these types, we focus on those for which $L(\lambda)$ and/or $R(\lambda)$ is a minimal basis, since they have favorable properties from the point of view of data compression and allow us to relate easily the degree of $P(\lambda)$ with some degree properties of the factors. We call these factorizations minimal rank factorizations. Motivated by the well-known fact that, generically, rank deficient polynomial matrices over the complex field do not have eigenvalues, we pay particular attention to the properties of the minimal rank factorizations of polynomial matrices without eigenvalues. We carefully analyze the degree properties of generic minimal rank factorizations in the set of complex $m \times n$ polynomial matrices with normal rank at most $r< \min \{m,n\}$ and degree at most $d$, and we prove that there are only $rd+1$ different classes of generic factorizations according to the degree properties of the factors and that all of them are of the form $L(\lambda) R(\lambda)$, where the degrees of the $r$ columns of $L(\lambda)$ differ at most by one, the degrees of the $r$ rows of $R(\lambda)$ differ at most by one, and, for each $i=1, \ldots, r$, the sum of the degrees of the $i$th column of $L(\lambda)$ and of the $i$th row of $R(\lambda)$ is equal to $d$. Finally, we show how these sets of polynomial matrices with generic factorizations are related to the sets of polynomial matrices with generic eigenstructures.

翻译：我们研究 $m \times n$ 多项式矩阵 $P(\lambda)$ 的秩揭示分解，将其分解为三个矩阵的乘积 $P(\lambda) = L(\lambda) E(\lambda) R(\lambda)$ 或两个矩阵的乘积 $P(\lambda) = L(\lambda) R(\lambda)$。在所有此类可能的分解中，我们重点关注那些 $L(\lambda)$ 和/或 $R(\lambda)$ 为最小基的分解，因为它们从数据压缩的角度具有优良性质，并且使我们能够轻松地将 $P(\lambda)$ 的次数与因子的某些次数性质联系起来。我们称此类分解为最小秩分解。受一个众所周知事实的启发，即在复数域上，秩亏缺的多项式矩阵一般没有特征值，我们特别关注无特征值的多项式矩阵的最小秩分解性质。我们仔细分析了在正规秩至多为 $r< \min \{m,n\}$、次数至多为 $d$ 的复数 $m \times n$ 多项式矩阵集合中，一般最小秩分解的次数性质，并证明根据因子的次数性质，仅存在 $rd+1$ 个不同的泛型分解类，且它们都具有 $L(\lambda) R(\lambda)$ 的形式，其中 $L(\lambda)$ 的 $r$ 列的次数至多相差 1，$R(\lambda)$ 的 $r$ 行的次数至多相差 1，并且对于每个 $i=1, \ldots, r$，$L(\lambda)$ 的第 $i$ 列的次数与 $R(\lambda)$ 的第 $i$ 行的次数之和等于 $d$。最后，我们展示了这些具有泛型分解的多项式矩阵集合与具有泛型特征结构的多项式矩阵集合之间的关系。