In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) \(A = (A_{\alpha \beta }x_{\alpha \beta })\), where \(A_{\alpha \beta }\) is a \(2 \times 2\) matrix over a field \(\mathbf {F}\) and \(x_{\alpha \beta }\) is an indeterminate for \(\alpha = 1,2,\dots , \mu \) and \(\beta = 1,2, \dots , \nu \). This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (SIAM J Matrix Anal Appl 16(3):719–734, 1995). Recent interests in this problem lie in the connection with non-commutative Edmonds’ problem by Ivanyos et al. (Comput Complex 27:561–593, 2018) and Garg et al. (Found. Comput. Math. 20:223–290, 2020), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial \(O((\mu \nu )^2 \min \{ \mu , \nu \})\)-time algorithm for computing the symbolic rank of a \((2 \times 2)\)-type generic partitioned matrix of size \(2\mu \times 2\nu \). Our algorithm is inspired by the Wong sequence algorithm by Ivanyos et al. for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of A for an arbitrary field \(\mathbf {F}\).

在本文中，我们考虑计算块结构符号矩阵（通用分区矩阵）的秩的问题\(A = (A_{\alpha \beta }x_{\alpha \beta })\)，其中\ (A_{\alpha \beta }\)是场\(\mathbf {F }\)上的\(2 \times 2\)矩阵，而\(x_{\alpha \beta }\)是\ (\alpha = 1,2,\dots, \mu \)和\(\beta = 1,2, \dots , \nu \). 这个问题可以看作是二分匹配问题的代数推广，岩田和室田 (SIAM J Matrix Anal Appl 16(3):719–734, 1995) 考虑了这个问题。最近对这个问题的兴趣在于与 Ivanyos 等人的非交换埃德蒙兹问题的联系。(Comput Complex 27:561–593, 2018) 和 Garg 等人。（Found. Comput. Math. 20:223–290, 2020），其中 Iwata 和 Murota 的结果隐含地指出，此类符号矩阵的秩和非交换秩 (nc-rank) 相同。本文的主要结果是一个简单的组合\(O((\mu \nu )^2 \min \{ \mu , \nu \})\) -time 算法，用于计算\(( 2 \times 2)\) - 大小为\(2\mu \times 2\nu \)类型的通用分区矩阵. 我们的算法受到 Ivanyos 等人的 Wong 序列算法的启发。用于一般符号矩阵的 nc 秩，并且不需要放大操作，不需要字段扩展，也不需要额外注意限制位大小。此外，它自然地为任意字段\(\mathbf {F}\)提供了A的最大秩完成。