On the Jacobi Matrix Inverse Eigenvalue Problem with Mixed Given Data

Jacobi Matrices (real symmetric tridiagonal matrices) have a wide range of applications in physics and engineering, and are closely and non-trivially linked with many other mathematical objects, such as orthogonal polynomial, one dimensional Schrödinger operators and the Sturm-Liouville problem. In the past couple of decades, constructing Jacobi matrices from different types of data was studied intensively. In this research, we show that a unique Jacobi matrix can be reconstructed from two types of mixed data: 1. its two eigenpairs, 2. its eigenvalues and the eigenvalues of its submatrix obtained by removing the first two rows and columns and one of its entry.