以下为句子列表：
英文: In this paper, through treating lines reciprocal transformation to a matrix, cogradiently reach the eigenvalue and eigenvector of a matrix, to solve the question treat a eigenvalue under without parameters, and given some advanced theorems.
中文: 摘要通过对矩阵进行行列互逆变换，同步求出矩阵特征值及特征向量，解决了不带参数求特征值问题，并给出一些新定理。
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英文: The relationship between the eigenvalue, eigenvector and the rank of the product of matrices AB and BA is discussed, and the factorization of the singular values of matrix A is obtained.
中文: 摘要讨论了矩阵乘积AB与BA的特征值、特征向量及秩等的关系，并得到了矩阵的奇异值分解。
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英文: Therefore, the selection of the optimum eigenvector of sample (regionalization index), sufficient representative samples, efficient and reasonable classifier and the deep understanding of the regions to be studied are the bases to guarantee the reasonable
中文: 为此，选取最优的样本特征向量（分区指标）、足够的有代表性的样本、高效合理的分类器以及对研究区域的深入了解将是保证分类结果合理可靠的基础。
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英文: We prove that 0 is the eigenvalue of the system's host operator, and finally we give the eigenvector of the eigenvalue 0.
中文: 并证明了0是系统主算子的本征值，给出了0本征值对应的本征向量。
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