On the construction of a Jacobi matrix from its mixed-type eigenpairs
The problem considered by the authors is to recover a Jacobi matrix \(J\) of finite size, \(n\times n,\) from a given set of eigenpairs associated with two principal submatrices. Observe that the Jacobi matrix \(J\) requires \(2n-1\) entries while the data used for the recovery is given by \(\lambda, \mu_1, \mu_2 \) and \[ \begin{aligned} {\mathbf Y}_1 &=(y_{1},y_{2},\dots,y_{k}) \\ {\mathbf {Y}_{2}} &=(y_{k+1},y_{k+2},\dots,y_{n}) \\ {\mathbf X} &=( x_{1},x_{2},\dots,x_{n})\end{aligned} \] where \(J{\mathbf X=}\lambda {\mathbf X}\), \(J_{i}{\mathbf {Y}_{i}}=\mu _{i} {\mathbf {Y}_{i}}\quad i=1,2\) and \(J_{i}\) are submatrices. After rescaling the vectors \({\mathbf Y}_{i}\), \({\mathbf X}\), the data reduces to \(2n\). The authors translate the inverse problem into a sequential algebraic system. They also manage, by using \(2\times 2\) determinants only, to find explicit conditions for the existence of a unique solution. The proof yields an algorithm for the reconstruction of the Jacobi matrix and numerical examples are provided in the last section.
scientific article; zbMATH DE number 1880074