一类矩阵迹方程正交解的一些研究

Some researches on orthogonal solutions to a class of matrix trace equations

应用正交矩阵标准形及其不变性得到了n阶矩阵迹方程■有正交解A=(alj)的充要条件,以及该方程的特征值都为实数或纯虚数的所有正交解的显示表达.由上述结果得到了相应迹方程的对称正交解的通解,并证明了其不存在反对称正交解.

By the canonical form of orthogonal matrix and its invariance, we obtain the necessary and sufficient conditions for the orthogonal solutions A=(aij) to an n order matrix trace equation■, and show the explicit expression of all general orthogonal solutions with the eigenvalues be all real or pure imaginary. Then we get the general symmetric orthogonal solutions to the corresponding trace equation, and prove that there is no antisymmetric orthogonal solution.