二阶方阵n次幂的普适公式及应用

摘要

二阶方阵的n次幂的研究一直没有给出完整的通用公式.采用矩阵变换及复数变换等研究方法,给出了二阶方阵n次幂的通用公式,给出了分式差分方程、二阶线性差分方程及差分方程组的完全解,应用该公式得到了二端梯形电阻网络等效电阻的通用公式.所得结果可使相关问题中的计算得到简化.%A complete universal formula of the n-th power of matrix,which has not been proposed in the literature ever before,is deduced in this study,by means of matrix transformation and complex representation.With the help of this formula,both the whole solutions for the difference equations with fractions,second order and linear difference equations and difference equation systems,and the universal formula for the equivalent resistance of laddered networks have been derived.Results obtained here can make calculation of the related problems easier.