On some Jacobi Matrix Inverse Eigenvalue Problems
with Mixed Given Data
CN-04
Yilin Yang
Advisor(s): Lei Cao
Undergraduate Student
College of Arts and Sciences
Mathematics
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 polynomials, 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 construct Jacobi matrices from two new types of data, and thus providing some new methods for solving inverse eigenvalue problem of Jacobi matrices. Using its special structure, we firstly show that a Jacobi matrix can be reconstructed from two eigenpairs by solving a nonhomogeneous linear system. Secondly, we show that a unique Jacobi matrix J could be determined by the eigenvalues of J, the eigenvalues of the submatrix obtained by removing the first two rows and columns from J and an arbitrary entry of J. Furthermore, we summarize equivalent sets of parameters used to recover Jacobi matrices and show some direct connections among these sets.
CN-04
Yilin Yang
Advisor(s): Lei Cao
Undergraduate Student
College of Arts and Sciences
Mathematics
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 polynomials, 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 construct Jacobi matrices from two new types of data, and thus providing some new methods for solving inverse eigenvalue problem of Jacobi matrices. Using its special structure, we firstly show that a Jacobi matrix can be reconstructed from two eigenpairs by solving a nonhomogeneous linear system. Secondly, we show that a unique Jacobi matrix J could be determined by the eigenvalues of J, the eigenvalues of the submatrix obtained by removing the first two rows and columns from J and an arbitrary entry of J. Furthermore, we summarize equivalent sets of parameters used to recover Jacobi matrices and show some direct connections among these sets.