Construction of a Jacobi matrix from spectral data (Q1239217)

A Jacobi matrix is defined to be a real, symmetric, tridiagonal matrix with positive off-diagonal entries. Given real numbers \(\{\omega_i\}\), \(i =1,2,\dots,n\) and \(\{\mu_j\}\), \(j=1,2,\dots,n-1\) which satisfy \[ \omega_1 < \mu_1 < \omega_2 < \mu_2 <\dots< \mu_{n-1} < \omega_n, \] it is shown that there exists a unique \(n \times n\) Jacobi matrix \(J\) such that \(\{\omega_i\}\) are the eigenvalues of \(J\) and \(\{\mu_j\}\) are the eigenvalues of the Jacobi matrix obtained by deleting the last row and column of \(J\). The uniqueness result was first demonstrated by Hochstadt.