Inverses, determinants, eigenvalues, and eigenvectors of real symmetric Toeplitz matrices with linearly increasing entries

A Note on the Eigensystem of the Covariance Matrix of Dichotomous Guttman Items

“…Elliott ( 1953 ) and Gregory and Karney ( 1969 ) identified that the eigenvalues λ i of S −1 , and (selected) corresponding eigenvectors P i , for i = 1, 2, …, n , are given by (in increasing order) and respectively (where m = 1, 2, …, n ). These results were later extended to more general tridiagonal matrices by Yueh ( 2005 ), see also Yueh and Cheng ( 2008 ) and Bünger ( 2014 ). Because the covariance matrix σ cov equals , the eigenvalues of σ cov are equal to times those of S , so they are also times the inverses of the eigenvalues of S −1 .…”

On special matrices related to Cauchy and Toeplitz matrices

“…In [3], the author obtained a unique LU factorizations and an explicit formula for the determinant and also the inversion of Toeplitz matrices. And, the inverse, determinants, eigenvalues, and eigenvectors of symmetric Toeplitz matrices over real number field with linearly increasing entries have been studied in [14]. In [15], the author showed that every n × n square matrix is generically a product of ⌊n/2⌋ + 1 and always a product of at most 2n + 5 Toeplitz matrices.…”

Reconstructing Latent Orderings by Spectral Clustering

“…Linearly decreasing Toeplitz matrices defined by A lin ij = b |i−j| = n − |i − j| have spectral properties analog to those of KMS matrices (trigonometric expression, interlacement, low frequency assigned to largest eigenvalue), but with more technical details available in Bünger (2014). This goes beyond the asymptotic case modeled by tridiagonal matrices.…”