论文标题
循环块矩阵的二进制等级
The Binary Rank of Circulant Block Matrices
论文作者
论文摘要
$ 0,1 $矩阵的二进制排名是其分区中最小的二进制等级，将其分为单色组合矩形。一个矩阵$ m $称为$（k_1，\ ldots，k_m; n_1，\ ldots，n_m）$ cricctant块对角线，如果是带有$ m $ m $对角块的块矩阵，则每个$ i \ in [m] $ th the $ i $ i $ m $ a $ $ m $ a $ $ m $ a $ a $ a $ $ m $ a $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ m $ $ n_i-k_i $零，所有其他条目都是零。在这项工作中，我们研究了这些矩阵及其补充的二进制等级。特别是，我们将这些矩阵的二进制等级与真实的等级进行了比较，后者对前者形成了下限。 我们提出了一种通用方法，可以证明具有某些指定结构和其他地方的块矩阵的二元级别上的上限。使用这种方法，我们证明了$（k_1，\ ldots，k_m; n_1，\ ldots，n_m）$循环的二进制等级，循环块diagonal矩阵的整数满足$ n_i> k_i> k_i> k_i> k_i> 0 $的二进制等级不超过[m]的最高等级。 $ \ gcd（n_i，k_i）-1 $上的所有$ i \ in [m] $。我们进一步为这些矩阵的二进制等级提供了几个足够的条件，以严格超过其实际等级。通过结合上限和下限，我们确定了各种矩阵家族的确切二进制等级，此外，还显着概括了Gregory的结果。 在铂尔曼的问题上，我们研究了$ k $ 0,1 $矩阵的二进制排名及其补充。作为我们在循环块对角线矩阵上的结果的应用，我们表明，对于每$ k \ geq 2 $，就存在$ k $ - g $ 0,1 $ 0,1 $矩阵，其二进制排名严格大于其补充。此外，我们准确地确定了每个整数$ r $，这是$ 2 $的$ 0,1 $ 0,1 $矩阵的最小二进制排名，其二进制等级$ r $。
The binary rank of a $0,1$ matrix is the smallest size of a partition of its ones into monochromatic combinatorial rectangles. A matrix $M$ is called $(k_1, \ldots, k_m ; n_1, \ldots, n_m)$ circulant block diagonal if it is a block matrix with $m$ diagonal blocks, such that for each $i \in [m]$, the $i$th diagonal block of $M$ is the circulant matrix whose first row has $k_i$ ones followed by $n_i-k_i$ zeros, and all of whose other entries are zeros. In this work, we study the binary rank of these matrices and of their complement. In particular, we compare the binary rank of these matrices to their rank over the reals, which forms a lower bound on the former. We present a general method for proving upper bounds on the binary rank of block matrices that have diagonal blocks of some specified structure and ones elsewhere. Using this method, we prove that the binary rank of the complement of a $(k_1, \ldots, k_m ; n_1, \ldots, n_m)$ circulant block diagonal matrix for integers satisfying $n_i>k_i>0$ for each $i \in [m]$ exceeds its real rank by no more than the maximum of $\gcd(n_i,k_i)-1$ over all $i \in [m]$. We further present several sufficient conditions for the binary rank of these matrices to strictly exceed their real rank. By combining the upper and lower bounds, we determine the exact binary rank of various families of matrices and, in addition, significantly generalize a result of Gregory. Motivated by a question of Pullman, we study the binary rank of $k$-regular $0,1$ matrices and of their complement. As an application of our results on circulant block diagonal matrices, we show that for every $k \geq 2$, there exist $k$-regular $0,1$ matrices whose binary rank is strictly larger than that of their complement. Furthermore, we exactly determine for every integer $r$, the smallest possible binary rank of the complement of a $2$-regular $0,1$ matrix with binary rank $r$.
