四元数实表示的代数应用

The Application of the Real Transformations with Quaternion for Algebra Theory

在四元数和四元数向量、矩阵空间上引入并交替使用三种不同的实数表示方式 ,将四元数体上的李雅普诺夫矩阵方程和二次型转换为实数域上的等价方程组和等价二次型 ,并在此基础上把四元数自共轭矩阵特征值、四元数向量和矩阵的常用范数、四元数矩阵的数值半径等运算问题一律转换为实数域上的等价运算问题 .

Three real transformations of quaternion vectors and matrixes were introduced. By using these transformations, Lyapunov matrix equation on quaternion field is changed to linear real equations, the quadric forms of quaternion will be transformed to the quadric forms of real, and the calculation about right eigenvalue of self-conjugate matrixes will be replaced by the eigenvalue of real matrixes. Some familiar norms of quaternion vector and matrix will be replaced by the familiar norms of real vector and matrix. The numerical radius of quaternion self-conjugate matrixes equals to a common norm of real matrixes.