中心对称矩阵的可逆性

Reversibility of center symmetric matrices

研究了中心对称矩阵的定义、结构及分块矩阵表示方法,利用分块矩阵的方法分别表示出偶数阶和奇数阶中心对称矩阵,以此为基础讨论偶数阶和奇数阶中心对称矩阵可逆的充分必要条件。找到对角相似分块矩阵,利用相似矩阵的性质得到偶数阶中心对称矩阵可逆性的充分必要条件。分别考虑了a=0和a≠0两种情况,得到了奇数阶中心对称矩阵可逆性的充分必要条件。研究了中心对称矩阵的逆矩阵求法公式,获得了一些新的结论,并结合一个具体例子说明了将阶数较高的中心对称矩阵的可逆性问题转化为阶数较低的矩阵的可逆性问题的方法,使大矩阵的运算化成小矩阵的运算,达到简化计算的目的,由所得结果可知中心对称矩阵的逆矩阵仍然是中心对称矩阵。

This article mainly studies the definition, structure and representation of the block matrix within centro symmetric matrices, and represents the centro symmetric matrices of even-order and odd-order utilized the block matrix method separately. Meanwhile, it discusses the sufficient and necessary condition of centro symmetric matrices of even-order and odd-order. The block matrix of diagonal similar be obtained, while using the property of the similar matrix, we can get the sufficient and necessary condition of reversibility with even-order centro symmetric matrix. Concerning the two kinds of conditions on a =0 and a #0, the sufficient and necessary condition of reversibility with odd-order centro symmetric matrix have also been received. Give the inverse matrices space formula of centro symmetric matrices, and obtain some new conclusions. Have a show on the method of reverse discussion that translates reverse questions of centro symmetric matrices with higher order into reverse questions of matrices with lower order, and make the operation of large matrix change into the operation of small matrix so as to reach the purpose of simplified calculation. Thereby we can know that the inverse matrix of eentro symmetric matrices is still centro symmetric matrices.