矩阵与其方幂相似的进一步讨论及应用

Further Discussion and Application on Matrix which is Similar to Its Power

证明了如果矩阵A的所有特征值都是1,则对任意非零整数k,A^k与A相似,并给出了A与A^k(k∈Z^+)相似的一些充分必要条件.作为应用,证明了一类非线性矩阵方程解的存在性.

For any nonzero integer k,a complex matrix A is similar to its power A^k if its all eigenvalues are 1.Furthermore,the necessary and sufficient conditions complex matrix A be similar to its power A^k( k∈Z^+) are given out.The existence of the solution to a class of nonlinear matrix equations is proved by applications.