矩阵方程A^m=J的某些解

On Some Solutions to Matrix Equation A^m = J

如果一个（０．ｌ）ｇ－循环矩阵的阶为ｃｋｍ，行和为ｃｋ且其 Ｈａｌｌ多项式被Ｔｃ（ｘ）Ｔｃ（ｘｃｎ）……Ｔｃ（ｘｃ（ｋ－１）ｍ）整除，其中ｍ，ｋ为正整数．ｃ为大于１的整数，Ｔｃ（ｘ）＝１＝ｘ＝……ｘ（ｃ－２），则它满足方程Ａｍ＝Ｊ，称之为这个方程的（ｃ，ｋ）－型解。本文用归纳法给出了某些（ｃ，ｘ）一型解的构造，并通过计算（ｃ，ｋ）－型解的秩．证明了方程Ａｍ＝Ｊ的不同型的解是不同构。最后，证明了方程Ａｍ＝Ｊ的所有（ｃ，１）－型解部是同构的．

If a (0, 1) g-circulant with the sum of its rows Ok is of order C^km, whose Hall-polynomial is divisible by Tc(x) Tc(xcm) ...Tc(xc(k-1)m), then it satisfies the matrix equation A^m=J, and is called a (c, k) -type solution to the equation, where m, k and c are positive integers with 0>1, Tc(x)=1+x+…+xc-1. Some (C, k)-tgpe solutions to the equation are constructed by induction. It is shown by calculating the ranks of (c, k)-type solutions, that solutions to the equation which are of different types are not isomorphic to each others Furthermore, all the (c, 1)-type solutions are mutually isomorphic.