求实矩阵指数的待定系数法的改进

Improvement of the Method of Undetermimed Coefficient for Exponential of a Real Matrix

在用待定系数法求矩阵指数ｅＡ时， 其中ｂｊ是方程组 （ｉ＝１，２，…，ｎ）的解，而λｉ是Ａ的特征值。本文证明了当Ａ为实矩阵，且λｉ≠λｊ（ｉ≠ｊ，ｉ，ｊ＝１，２，…，ｎ），时，ｂｊ是唯一的且是实数；并进一步证明了，任给复域上的共轭复数λｉ，且λｉ≠λｊ（ｉ≠ｊ，ｉ，ｊ＝１，２，…，ｎ），上述方程组的解ｂｊ仍是实数。

When an exponential of a matrix eA is solved for by the Method of undetermined coefficient, eA, Where (b0b1…n-1)T is the solution to the group of equations eλi=(i=1,2…,n) with λi being eigenvalues of the matix A. It is presented that when the matric A is real and λi≠λj (i≠j,i,j=1,2,…,n),bj is only and real. And it is also presented that when λis are conjugste and λi≠λj(i≠j,i,j=1, 2,…,n),the solution to the previous group of equations (b0b1…bn-1)T is still real.