三个重要矩阵秩的不等式证明
Proofs of Three Important Matrix Inequalities

在高等代数的广阔领域内，矩阵作为核心研究对象，其理论贯穿于学科始终。矩阵的秩，作为矩阵的核心属性，其性质与结论的研究至关重要。矩阵秩有许多重要的不等式，当然它们的证明方法也有很多，而本文中的三个不等式在很多课本中没有给出证明。本文聚焦于矩阵秩的三个关键不等式，它们反映了矩阵的和、乘积的秩与原矩阵秩之间的关系，主要从矩阵向量组的秩线性相关性及齐次线性方程组的相关性质等方面证明有关矩阵秩的三个重要不等式。
In the broad field of higher algebra, matrix is the core research object, and its theory runs through the whole subject. As the core attribute of matrix, the rank of matrix is very important to study its properties and conclusions. There are many important inequalities for matrix rank, and of course there are many ways to prove them. This paper focuses on the three key inequalities of matrix rank, and proves the three important inequalities of matrix rank from the linear correlation of matrix rank and the correlation properties of homogeneous linear equations.