标题： 带有应用的棋盘图案矩阵的秩信息嵌入和检索 显示英文标题

标题： Ranks of Checkered Pattern Matrices with Applications to Information Embedding and Retrieving

摘要： 棋盘图案的特点是其正方形结构以及仅使用两种不同的颜色。 这些颜色通常由两种数值集合表示： {1,0} 和 {1,-1}。 基于 {1,0}的矩阵在形成棋盘图案时可能与基于 {1,-1}的矩阵看起来相同，因为唯一的区别是将数字 0 改为 -1。 然而，从数学角度来看，这两种矩阵是完全不同的，因为使用 {1,0}的矩阵的秩为 2，而使用 {1,-1}的矩阵的秩为 1。 提前了解这一差异可以使我们在进行信息嵌入和提取等矩阵操作时减少计算工作量。 显示英文摘要

摘要： Checkered patterns are characterized by their square structure and the use of only two distinct colors. These colors are typically represented by two types of numerical sets: {1,0} and {1,-1}. Matrices based on {1,0} may seem identical to those based on {1,-1} when forming checkered patterns because the only difference is that the numbers 0 are changed to -1. However, these two kinds of matrices are completely different in a mathematical sense because a matrix using {1,0} has a rank of 2 and a matrix using {1,-1} has a rank of 1. Knowing this difference in advance allows us to reduce the computational effort required for matrix operations such as information embedding and retrieving.