非线性矩阵方程的一般解(英文)

The general solution of Non-linear Matrix Equation

与研究m次代数方程相类似地研究了m次非线性矩阵方程,给出了一般解的求法,一般解的结构,一般解的线性组合的性质.当矩阵是非奇异矩阵时,它的m次矩阵根是有限个,特别是一个非奇异的Jordan块的m次矩阵根有m个.当矩阵是奇异矩阵时,它可能有m次矩阵根,也可能没有m次矩阵根,这由它的特征值及对应的Jordan块阶数决定.这种判定方法又直接导出了m次可解矩阵方程根的公式,及非奇异矩阵的m次根的表达式.最后我们也在各种不同的情况给出了结论的数值例子.

The general solution of solvable matrix equation X^m=A was obtained. If A is non-singular matrix, the equation X^m=A has finitely many solutions. Specially, the number of the m-root of non-singular Jordan block was m. When the matrix is a singular matrix, it may have m-root, also possible have no m-root. This is determined by their Jordan normal form. And when m is an even number, the solutions of X^m=A span whole Cn×n if and only if A=λI with λ≠0. We also determine the precise number of solutions in various cases.