一种三对角矩阵的相似对角化及其应用  

Similarity Diagonalization of a Tridiagonal Matrix and its Applications

作　　者：张兴刚 曹磊 ZHANG Xinggang;CAO Lei(School of Physics,Guizhou University,Guiyang 550025,China;Powerchina Guiyang Engineering Co.Ltd.,Guiyang 550081,China)

机构地区：[1]贵州大学物理学院,贵州贵阳550025 [2]中国电建集团贵阳勘测设计研究院,贵州贵阳550081

出　　处：《贵州大学学报（自然科学版）》2021年第6期19-24,共6页Journal of Guizhou University:Natural Sciences

基　　金：国家自然科学基金资助项目(11965007);贵州省科技合作计划资助项目(20157641)。

摘　　要：针对二维筒仓中随机堆积的颗粒体系,建立了一个由吸收系数p和侧向传递系数q决定的格点系统模型。导出了筒仓底部的压力分布与顶部压力分布的关系式,该关系式的解析计算涉及到三对角形式的传递系数矩阵A(p,q)的相似对角化。通过相似变换将A(p,q)对称化,并且提出基于二阶差分方程的方法讨论这种对称三对角矩阵的特征值和特征向量,只有在特殊情况下才能解析地求出A(p,q)的所有特征值并将其相似对角化。最后将相关的数学结果用于计算有效质量随颗粒总质量的变化关系,结果表明格点系统模型可以得到与Janssen模型类似的结果。A lattice model which is determined by the absorption coefficient p and the lateral transfer coefficient q is proposed for the two dimensional disordered granular columns.The pressure distribution relation between the top and the bottom of the container is deduced,and the analytic calculation of this mathematical relation involves the similarity diagonalization of tridiagonal transfer matrix A(p,q).A similarity transformation is proposed to symmetrize A(p,q),and the method based on the second order difference equations is proposed to obtain the eigenvalues and eigenvectors of the symmetric matrix.Finally,these mathematical results are used to discuss Janssen effect.Calculations show that results similar to Janssen model can be produced by our lattice model.

关 键 词：粮仓效应 格点模型 三对角矩阵 特征值 特征向量 

分 类 号：O151[理学—数学] O312[理学—基础数学]