Asymptotic distribution of the even and odd spectra of real symmetric Toeplitz matrices Original Research Article

If Tn=(tr−s)r,s=0n is a real symmetric Toeplitz (RST) matrix then image has a basis consisting of left ceilingn/2right ceiling eigenvectors x satisfying (A) Jx=x and left floorn/2right floor eigenvectors y satisfying (B) Jy=−y, where J is the flip matrix. We say that an eigenvalue λ of Tn is even if a λ-eigenvector of Tn satisfies (A), or odd if a λ-eigenvector of Tn satisfies (B). We call the collection of even (odd) eigenvalues of Tn the even (odd) spectrum of Tn. In the case where tr=1/π ∫0πf(x) cosrx dx a great deal is known about the asymptotic distribution of the eigenvalues of Tn as n→∞, under suitable assumptions on f. However, the question of the separate asymptotic distributions of the even and odd spectra does not seem to have been raised. This is the subject of this paper.