An Inverse Eigenvalue Problem for Jacobi Matrices

Abstract

Let $T_{1,n}$ be an $n\times n$ unreduced symmetric tridiagonal matrix with eigenvalues $$\lambda_1<\lambda_2<\cdots<\lambda_n.$$ and $$W_k=\Bigg(\begin{matrix}T_{1,k-1} & 0  \\0&T_{k+1,n}\end{matrix} \Bigg)$$is an $(n-1)\times(n-1)$ submatrix by deleting the $k^{th}$ row and $k^{th}$ column, $k=1,2,\ldots,n$ from $T_n$. Let $$\mu_1\leq\mu_2\leq\cdots\leq\mu_{k-1}$$ be the eigenvalues of $T_{1,k-1}$ and $$\mu_k\leq\mu_{k+1}\leq\cdots\leq\mu_{n-1}$$ be the eigenvalues of $T_{k+1,n}$.

A new inverse eigenvalues problem has put forward as follows: How do we construct an unreduced symmetric tridiagonal matrix $T_{1,n}$, if we only know the spectral data: the eigenvalues of $T_{1,n}$, the eigenvalues of $T_{1,k-1}$ and the eigenvalues of $T_{k+1,n}$?
Namely if we only know the data: $\lambda_1,\lambda_2,\cdots,\lambda_n,\mu_1,\mu_2,\cdots,\mu_{k-1}$ and $\mu_k,\mu_{k+1},\cdots,\mu_{n-1}$ how do we find the matrix $T_{1,n}$? A necessary and sufficient condition and an algorithm of solving such problem, are given in this paper.