论文标题
重新访问矩阵多项式最大的常见除数
Revisiting the matrix polynomial greatest common divisor
论文作者
论文摘要
In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(λ)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column dimension $n$.我们提供了矩阵$ g（\ la）\ in \ f [\ la]^{\ ell \ times n} $的必要条件，使用$ m \ times n $ n $复合矩阵$ p（λ）的史密斯正常形式，将其作为gcrd，是通过将$ p_i（λ）$ pertionaly $ _ i = 1 i = 1 i = 1 i = 1 i = 1我们还描述了解决方案$ g（\ la）$的完整自由度，并将其链接到$ p（\ la）$的史密斯表格和Hermite形式。然后，当使用状态空间技术（$ \ f = \ c $或$ \ r $）时，我们给出了用于为此问题构建特定最低等级解决方案的算法。此新方法仅使用正交转换直接适用于$ p（\ la）$的系数矩阵。该方法基于楼梯算法，该算法应用于$ p（\ la）$的广义状态空间模型的特定铅笔。
In this paper we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices $P_i(λ)\in \F[\la]^{m_i\times n}$, $i=1,\ldots,k$ with coefficients in a generic field $\F$, and with common column dimension $n$. We give necessary and sufficient conditions for a matrix $G(\la)\in \F[\la]^{\ell\times n}$ to be a GCRD using the Smith normal form of the $m \times n$ compound matrix $P(λ)$ obtained by concatenating $P_i(λ)$ vertically, where $m=\sum_{i=1}^k m_i$. We also describe the complete set of degrees of freedom for the solution $G(\la)$, and we link it to the Smith form and Hermite form of $P(\la)$. We then give an algorithm for constructing a particular minimum rank solution for this problem when $\F=\C$ or $\R$, using state-space techniques. This new method works directly on the coefficient matrices of $P(\la)$, using orthogonal transformations only. The method is based on the staircase algorithm, applied to a particular pencil derived from a generalized state-space model of $P(\la)$.
