求解具有重特征值的线性四元数微分方程
Solve the linear quaternion-valued differential equations having multiple eigenvalues
摘要：
二维线性四元数值微分方程（QDEs）的理论是最近建立的（见Kou和Xia，SAPM）。QDE和ODE之间的一些深刻的差异被观察到。此外，在[Kou and Xia，SAPM]中，还提出了一种利用特征值和特征向量计算基本矩阵的算法。然而，只要特征值是简单的，就可以构造基本矩阵。如果线性系有多个特征值，如何构造基本矩阵？特别地，如果独立特征向量的数目可能小于系统的维数。也就是说，特征向量的数目不足以构造基本矩阵。如何找到“缺失的解决方案”？本文的主要目的就是回答这个问题。[Kou and Xia，SAPM]中利用四元数值矩阵的Caley行列式对QDE进行了进一步的研究。Caley行列式的一个很大的缺点是它可以沿着不同的行和列展开。由于四元数的非交换性，这可能导致不同的结果。这种方法不便于使用。给出了基于置换的四元数矩阵行列式的新定义，并对该定义进行了理论分析。这种新的行列式定义与Calay行列式相比有很大的优越性。
英文摘要：
The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established (see Kou and Xia, SAPM). Some profound differences between QDEs and ODEs were observed. Also, an algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented in [Kou and Xia, SAPM]. However, the fundamental matrix can be constructed providing that the eigenvalues are simple. If the linear system has multiple eigenvalues, how to construct the fundamental matrix? In particular, if the number of independent eigenvectors might be less than the dimension of the system. That is, the numbers of the eigenvectors is not enough to construct a fundamental matrix. How to find the "missing solutions"? The main purpose of this paper is to answer this question. Furthermore, Caley determinant for Quaternion-valued matrix was adopted to proceed the theory of QDEs in [Kou and Xia, SAPM]. One big disadvantage of Caley determinant is that it can be expanded along the different rows and columns. This may lead to different results due to non-commutativity of the quaternions. This approach is not convenient to be used. The novel definition of determinant for Quaternion-valued matrix based on permutation is introduced to analyze the theory. This newly definition of determinant has great advantage compare to Calay determinant.
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