四元数

Basing on the famous Schur theorem on real quaternion division algebra and the operation of matrix, some properties and judging criterions of skew self-conjugate matrix are obtained. Several theorems about characteristic value, similar decomposition and real expression of skew self- conjugate matrix are gained.

借助四元数体上的Schur三角分解定理和体上矩阵的运算，得到了斜自共轭矩阵的一些性质及判定准则，获得了斜自共轭矩阵的实表示、相似分解以及特征值的几个定理。

Then it simplifies the proof of division ring of quaternion .

从而简化了四元数作为除环的证明。

Simulation shows the algorithm validity and the attitude motion equation in error quaternion are effective to omnidirectional attitude control.

仿真表明了全角度算法的正确性，以及基于文中的四元数姿态误差方程可有效进行导弹全方位姿态控制。

Known, the Euler angles and quaternion is a correlation between them.

众所周知，欧拉角和四元数是存在对应关系的。

Three dimensional rotation represented by quaternion has two advantages over that by matrix: first, the geometric meaning is definitude; second, computation is easy.

用四元数表达三维的旋转与使用矩阵相比具有两个优点：第一，几何意义明确；第二，计算简单。

Quaternions do not suffer from gimbal lock, because they do not express rotations relative to three separate axes.

四元数就不会遭受万向节锁的痛苦，因为它不用三个分离的轴表示旋转。

And find that each element in the group can be decomposed into the product of two quaternion component matrices.

指出SO(4)群中每个矩阵都可以分解为两种单位四元数分量矩阵乘积的形式。

Zeros of irreducible real quaternionic polynomial are the isolated point solutions of the original polynomial equation.

不可约四元数多项式的零点对应原多项式的孤立点解，一次实系数多项式的零点对应原多项式的实数解，二次实系数多项式(Δ＜0)的零点对应原多项式的等价类解。

We know that examples of special Lagrangian submanifolds are very important for studying these submanifolds.

我们也采取相应的思想，得到一些四元数欧式空间中有关极小拉格朗日的伯恩斯坦型定理。

In this note, we show that for any two matrices A and B over a generalized quaternion algebra defined on an arbitrary field F of characteristic not equal to two, if A and B are similar and the main diagonal elements of A and B are in F, then their traces are equal.

本文对于特征不是2的任意域F上定义的广义四元数代数上的两个矩阵A和B，给出如果A和B相似并且它们的主对角线上的元素在F中，那么它们的迹相等。

But we don't care about Battlegrounds.

但我们并不在乎沙场中的显露。

Ah! don't mention it, the butcher's shop is a horror.

啊！不用提了。提到肉，真是糟透了。