Constructing Jacobi Matrices from Mixed Data
Yilin Yang August 13, 2014
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, the Hamburger moments problem and Sturm-Liouville problem. In the past couple of decades, constructing Jacobi matrices from different types of data was studied intensively. In this paper, we construct Jacobi matrices from a more general data type than considered before. By using the special structure of Jacobi matrix and its eigenpairs, we successfully find a method to recover Jacobi matrices from two eigenpairs by solving a nonhomogeneous linear system and show that the two eigenvalues can be replaced by any two elements of the Jacobi matrices. Meanwhile, we give the conditions of uniqueness and existence of the solution. Furthermore, we prove that different sets of parameters used to recover Jacobi matrices by other authors are equivalent by exploring the direct connections between these different types of mixed data without necessarily recovering the matrices. Also a MATLAB code was developed to compute the Jacobi matrices based on our method.
Yilin Yang August 13, 2014
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, the Hamburger moments problem and Sturm-Liouville problem. In the past couple of decades, constructing Jacobi matrices from different types of data was studied intensively. In this paper, we construct Jacobi matrices from a more general data type than considered before. By using the special structure of Jacobi matrix and its eigenpairs, we successfully find a method to recover Jacobi matrices from two eigenpairs by solving a nonhomogeneous linear system and show that the two eigenvalues can be replaced by any two elements of the Jacobi matrices. Meanwhile, we give the conditions of uniqueness and existence of the solution. Furthermore, we prove that different sets of parameters used to recover Jacobi matrices by other authors are equivalent by exploring the direct connections between these different types of mixed data without necessarily recovering the matrices. Also a MATLAB code was developed to compute the Jacobi matrices based on our method.