论文标题
针对四个矩阵的特征向量 - eigenvalue及其算法和计算机程序的身份
The eigenvector-eigenvalue identity for the quaternion matrix with its algorithm and computer program
论文作者
论文摘要
Peter Denton, Stephen Parke, Terence Tao and Xining Zhang [arxiv 2019] presented a basic and important identity in linear commutative algebra, so-called {\bf the eigenvector-eigenvalue identity} (formally named in [BAMS, 2021]), which is a convenient and powerful tool to succinctly determine eigenvectors from eigenvalues.该身份将特征向量的组件与$ a $和未成年$ m_j $的特征值联系起来，该$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ i}^{n-1}（{λ_i}（a） - {λ_k}（a））= \ prod_ {k = 1}^{n-1}（{λ_i}（a） - {a） - {λ_k}（m_j）（m_j））。 \，\，\，％\ mbox {（\ cite {tao-eig，d-p-t-z}）} \]} \] 实际上，它已被广泛应用于各个领域，例如数值线性代数，随机矩阵理论，逆特征值问题，图理论，中微子物理学等。在本文中，我们将特征向量 - 元素值身份扩展到了非交通性的季节划分环。建立了针对四个矩阵的特征向量 - 元素值身份。此外，我们给出了一种新方法和算法，以从右征值的右征值中计算特征向量。一个程序旨在实现算法来计算特征向量。一个开放的问题结束了纸张。一些示例显示了该算法和程序的良好性能。
Peter Denton, Stephen Parke, Terence Tao and Xining Zhang [arxiv 2019] presented a basic and important identity in linear commutative algebra, so-called {\bf the eigenvector-eigenvalue identity} (formally named in [BAMS, 2021]), which is a convenient and powerful tool to succinctly determine eigenvectors from eigenvalues. The identity relates the eigenvector component to the eigenvalues of $A$ and the minor $M_j$, which is formulated in an elegant form as \[ \lvert v_{i,j} \rvert^2\prod_{k=1;k\ne i}^{n-1}({λ_i}(A)-{λ_k}(A))=\prod_{k=1}^{n-1}({λ_i}(A)-{λ_k}(M_j)). \,\,\,%\mbox{(\cite{tao-eig,D-P-T-Z})} \] In fact, it has been widely applied in various fields such as numerical linear algebra, random matrix theory, inverse eigenvalue problem, graph theory, neutrino physics and so on. In this paper, we extend the eigenvector-eigenvalue identity to the quaternion division ring, which is non-commutative. A version of eigenvector-eigenvalue identity for the quaternion matrix is established. Furthermore, we give a new method and algorithm to compute the eigenvectors from the right eigenvalues for the quaternion Hermitian matrix. A program is designed to realize the algorithm to compute the eigenvectors. An open problem ends the paper. Some examples show a good performance of the algorithm and the program.
