德州学院学报
德州學院學報
덕주학원학보
JOURNAL OF DEZHOU UNIVERSITY
2011年
6期
1-4
,共4页
矩阵秩%矩阵多项式%解空间%特征多项式%最小多项式
矩陣秩%矩陣多項式%解空間%特徵多項式%最小多項式
구진질%구진다항식%해공간%특정다항식%최소다항식
matrix rank%matrix polynomial%solution space%characteristic polynomial%minimal polynomial
借助以矩阵多项式为系数矩阵的齐次线性方程组解空间的直和分解结果，给出了一般数域上矩阵多项式秩的几个基本恒等式．作为应用，得到了复数域上矩阵可对角化的一个充要条件，给出了复数域上线性空间关于其上的线性变换的准素分解定理的简洁证明．最后提出一个关于矩阵多项式秩等式的公开问题．
藉助以矩陣多項式為繫數矩陣的齊次線性方程組解空間的直和分解結果，給齣瞭一般數域上矩陣多項式秩的幾箇基本恆等式．作為應用，得到瞭複數域上矩陣可對角化的一箇充要條件，給齣瞭複數域上線性空間關于其上的線性變換的準素分解定理的簡潔證明．最後提齣一箇關于矩陣多項式秩等式的公開問題．
차조이구진다항식위계수구진적제차선성방정조해공간적직화분해결과，급출료일반수역상구진다항식질적궤개기본항등식．작위응용，득도료복수역상구진가대각화적일개충요조건，급출료복수역상선성공간관우기상적선성변환적준소분해정리적간길증명．최후제출일개관우구진다항식질등식적공개문제．
Based on the results of the direct sum decomposition on the solution space of homogeneous linear equations which coefficient matrix is matrix polynomial, several identities of the rank of matrix polynomi als on general number field are presented. As applications, a sufficient and necessary condition is derived to diagonalizable of matrix on complex number field, and a concise proof is presented to primary decompo sition theorem of the linear space on complex number field with respect to the linear transformation on it. Finally, an open question of the rank identity of matrix polynomials is brought up.
