发表机构
CUNEF Universidad（CUNEF大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有理函数域上的矩阵，通过特殊化方法，仅用线性代数求解 Penrose 条件，并精确刻画伪逆失效的参数值（即秩下降点），给出符号算法并验证于经济模型。
AI 中文摘要
本文考虑元素为多个参数的有理函数、取值于 Moore-Penrose 域（即具有对合自同构且每个矩阵都有 Moore-Penrose 逆的域）的矩阵。我们证明，在任意这样的域上，Penrose 条件虽然构成一个二次多项式系统，但仅用线性代数即可求解。随后，我们用元素的次数和秩（而非矩阵的维数）来界定伪逆的分子和分母的次数。特殊化是主线：闭式形式在有理函数域上计算一次，问题在于哪些参数值下它仍然返回特殊化矩阵的伪逆。我们仅从矩阵本身、在计算伪逆之前，就确定它不成立的参数值。这些值是由最大子式构造的多项式的零点，也是秩下降的参数值。这将之前的充分条件转化为精确刻画。该理论还给出了实数、复数和参数情形的符号算法，我们在 Maple 中实现，并在 972 次计时重复实验上进行了测试。我们还将它们应用于 Leontief 经济模型，其中被排除的值是经济停止可行的临界点。
英文摘要
In this paper we consider matrices whose entries are rational functions of several parameters over a Moore-Penrose field, that is, a field with an involutory automorphism over which every matrix has Moore-Penrose inverse. We prove that over any such field the Penrose conditions can be solved with linear algebra alone, even though they form a polynomial system of degree two. We then bound the degrees of the numerator and of the denominator of the pseudoinverse in terms of the degree of the entries and of the rank, and not of the dimensions of the matrix. Specialization is the thread: the closed form is computed once over the field of rational functions, and the question is at which parameter values it still returns the pseudoinverse of the specialized matrix. We determine the values where it does not, from the matrix alone and before computing the pseudoinverse. They are the zeros of a polynomial built from the maximal minors, and they are the values at which the rank decreases. This turns previous sufficient conditions into an exact characterization. The theory also gives symbolic algorithms for the real, the complex and the parametric case, which we implement in Maple and test on 972 timed replications. We also apply them to a Leontief economic model, where the excluded value is the point at which the economy stops being viable.
Comments25 pages, 1 figure, 6 tables. Replication package: https://doi.org/10.5281/zenodo.23164326
