零和自由半环上的半可逆矩阵

Semi-invertible Matrices over Zero-sum-free Semirings

在零和自由半环上,举例说明矩阵方程组AX=B和X+A_1B=A2B并不是在所有情况下都同解,其中A是已知的n×n阶半可逆矩阵,X是未知的n维列向量,A_1和A_2分别满足条件I+AA_1=AA_2和I+A_1A=A_2A.得到关于方程AX=B和X+A_1B=A_2B同解的一些条件,完善零和自由半环上半可逆矩阵的相关性质,扩展矩阵的应用范围.

Over zero-sum-free semirings, we give an example to show that matrix equations AX = B andX ＋ A 1B = A 2B do not al- ways have the same solutions, where A is a known n x n semi-invertible matrix and B is an unknown n-dimensions column vector, A i and A z satisfy I ＋ AA 1 = AA 2 and I ＋ A1 A = A 2A. We present some conditions under which the systems AX = B and X ＋ A 1B = A 2B have the same solutions and give some properties of semi-invertible matrices. Our results extend the scope of the application of matrices.