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.%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．
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