Eigenvalues and eigenvectors of some tridiagonal matrices
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.
scientific article; zbMATH DE number 149642