Eigenvalues and eigenvectors of some tridiagonal matrices
type: Item
instance of: scholarly article
title: Eigenvalues and eigenvectors of some tridiagonal matrices
zbMATH Open document ID: 0771.15004
DOI: 10.1007/BF00051649
published in: Acta Mathematica Hungarica
publication date: 1993-04-01T00:00:00Z
review text: If \(\nu\), \(a\), \(b\), \(s\), \(t\) are complex numbers the \((n+1)\times(n+1)\) matrix \(M_{n,k}\) is the tridiagonal matrix whose elements \(m_{jl}\) \((j,l=0,1,\dots,n)\) are given by \[ m_{jj}=\begin{cases} a+\nu &\text{ if } j=0,\dots,k-1\\\nu &\text{ if } j=k,\dots,n-k\qquad (n+1-2k\geq 0),\\ b+\nu &\text{ if } j=n-k+1,\dots,n\end{cases} \] \[ m_{jj}=\begin{cases} a+\nu &\text{ if } j=0,\dots,n-k\\ a+b+\nu &\text{ if } j=n-k+1,\dots,k-1\qquad (n+k-2k<0)\\ b+\nu &\text{ if } j=k,\dots,n\end{cases} \] and \[ m_{jl}=\begin{cases} s &\text{ if } j=l+k,\;l=0,\dots,n-k\\ 0 &\text{ if } 0<| j-l|\neq k\\ t &\text{ if } l=j+k,\;j=0,\dots,n-k\end{cases} \] The author derives expressions for the determinant, the eigenvalues and the eigenvectors of \(M_{n,k}\) and he derives new quadratic inequalities of Wirtinger type.
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 149642
zbMATH Keywords: tridiagonal matrix
zbMATH Keywords: determinant
zbMATH Keywords: eigenvalues
zbMATH Keywords: eigenvectors
zbMATH Keywords: quadratic inequalities
author: László Losonczi
reviewed by: Fergus J. Gaines
MaRDI profile type: Publication
cites work: Discrete analogs of inequalities of Wirtinger
cites work: Q4775958
cites work: Q3781298
cites work: On discrete inequalities of Wirtinger's type
cites work: Q5795611
cites work: Q5785756
description: scientific article; zbMATH DE number 149642