# Inverse eigenvalue problems for checkerboard Toeplitz matrices

Research paper by T H Jones, N B Willms

Indexed on: 08 Jul '18Published on: 03 Jul '18Published in: Journal of physics. Conference series

#### Abstract

The inverse eigenvalue problem for real symmetric Toeplitz matrices motivates this investigation. The existence of solutions is known, but the proof, due to H. Landau, is not constructive. Thus a restriction, namely the required eigenvalues are to be equally spaced, is considered here. Two types of structured matrices arise, herein termed, “checkerboard” and “outer-banded”. Examples are presented. Properties of these structured matrices are explored and a full characterization of checkerboard matrices is given. The inverse eigenvalue problem is solved within the class of odd checkerboard matrices. In addition the “symmetric-spectrum” inverse eigenvalue problem is solved within a subclass of Hankel matrices. A regularity conjecture of H. Landau for the Toeplitz inverse eigenvalue problem is discussed and a similar conjecture for checkerboard Toeplitz matrices is given.