Inverse eigenvalue problems for checkerboard Toeplitz matrices

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.