【摘要】 为刻画控制理论领域涉及较多的二次矩阵方程的解,利用固定点理论和矩阵范数的相关知识,给出该类方程有解的充分必要条件和有唯一解的判定定理.得出矩阵方程X2-2AX+B=0(A,B,X是n×n矩阵)至少有一个解矩阵的充分条件是2n阶构造矩阵R的特征值是两两不同的.并且当‖A‖≤1,‖B‖<1/2时,矩阵方程AX2-2X+B=0在闭球B‖B‖(0)上有唯一的解.更多还原

【Abstract】 We studty the general quadratic matrix equation which has been applied to control theory and graph theory.The sufficient and essential condition of the solution and the judgment theorem of unique solution were obtained by fixed-point theory and matrix norm.Let A,B,X denote n×n matrices,a sufficient condition for the existence of at least one solution of matrix equation X2-2AX+B=0 is that eigenvalues of the 2n×2n matrix R be pairwise different.When ‖A‖≤1,‖B‖<12,the matrix equation AX2-2X+B=0 has an unique solution in the closed ball B‖B‖(0).更多还原