来自双对角分解的矩阵连分数
Positive Bidiagonal Factorizations: Combinatorial Structure, Spectral Theory, and Matrix Continued Fractions
- Complutense University of Madrid（马德里康普顿斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文从有序双对角分解构造矩阵连分数，通过循环线性化和有理高度排序转化为块三对角形式，截断得到 Padé 型逼近，并给出收敛性、残差估计及多种谱实现。
AI中文摘要：
本文从有序双对角分解 $T=L_1\cdots L_pU_q\cdots U_1$ 出发，构造了一个矩阵连分数，其系数为各个双对角参数。一种循环线性化将 $T$ 的一次作用分解为 $p+q$ 个基本步骤，并通过有理高度排序将所得算子转化为块三对角形式。在高度块 $N$ 处截断，得到一个结构化的 Padé 型逼近，该逼近与返回级数在 $2N+1$ 次以内一致。其分母行列式分解为连续 Schur 补主元的行列式之积。文中识别了首个被省略的系数，并且算子和有限截面假设给出了带可计算残差估计的收敛性。对于非负系数，收敛项在谱半径圆盘内的正实区间上逐项单调，且其分母为非奇异 $M$-矩阵。在正混合 Favard 假设下，矩形投影产生 $q\times p$ Weyl 矩阵的 Padé 型逼近。当 $q=1$ 时，该构造退化为 $p$ 分支 Stieltjes 分数；当 $p,q>1$ 时，其尾部为矩阵值。文中发展了两种显式谱实现。混合 Piñeiro 系统提供了闭式因子系数以及一个有理 $(3,2)$ 测试，用于验证收敛性和残差估计。类似 Jacobi 的扩展产生了第二族矩形 Weyl 矩阵，并且完全 Gamma 因子相消恢复了 Piñeiro 情形。
英文摘要：
Prescribed positive bidiagonal factorizations of semi-infinite banded matrices reveal spectral, approximation, and integrable structures for arbitrary lower and upper bandwidths. Every cyclic permutation of the factors admits a normalized lower--upper positive factorization. Under suitable degree conditions, factor transfers induce matrix Christoffel transformations, linking the cyclic Darboux orbit to mixed-type Favard theory. Christoffel words determine minimum-height positive refactorizations and the cyclic products that reach them. The associated matrices have two block displacements, reducing to two diagonals for coprime bandwidths. Their finite nonzero spectra lie on a star, and planar networks yield positive radial Stieltjes moment sequences. In the compact radial case star-supported representing measures are characterized by conditions at the origin and negative fractional moments. For coprime bandwidths, a radial moment deformation gives determinant solutions of a sparse Lax hierarchy and synchronizes the cyclic Toda flows. Retaining the prescribed factor order yields matrix continued fractions with explicit Padé-type contact, denominator factorizations, backward evaluation, and error bounds. For bounded nonnegative factors, the convergents are monotone and converge to the least nonnegative solution in the admissible domain. Under the Favard identification, they approximate the mixed-type Weyl matrix. Unbounded factorizations retain the formal approximation; analytic convergence requires a closed realization and stability hypotheses, with an additional identification for measure-defined Weyl matrices. Piñeiro and Jacobi-like systems give applications, including global positive coefficientwise integrable solutions under the stated positivity and AT hypotheses.
