关于2×2分块矩阵秩扰动的界

Upper and Lower Bounds for the Ranks of Perturbation Matrices on a Class of 2×2 Matrices

研究一类2×2分块矩阵秩的不等式以及一个含参变量X的矩阵函数秩的等价条件。利用矩阵Schur补的技巧及矩阵广义逆的特性,首先给出2×2分块矩阵M=(ACB0)秩的结果,随后讨论了在一定条件下2×2分块矩阵M_X=(ACBX)秩的上下界,最后获得矩阵函数A-BXC的秩与X选取无关的等价条件。

The inequalities of the ranks on a kind of 2 × 2 partitioned matrices and the equivalent conditions on ranks of matrix function are researched.Firstly, the result of the rank of the matrix M=（C^A O^B） is obtained by using techniques of the Schur complement and the properties of generalized inverse of matrix. Secondly, the upper and lower bounds for ranks of the perturbation matrix Mx=（C^A X^B） are studied under certain conditions. Finally, equivalent conditions are given when the rank of A -BXC and X are irrelevant.