Numerical Study of Spectral Behavior of the Grcar Matrix

The Grcar matrix is a class of non-symmetric Toeplitz upper Hessenberg matrices with banded structure. In a recent paper [G. Meurant, A note on the Grcar Matrix, Numerical Algorithms, https://doi.org/10.1007/s11075-024-01999-2], the recursive formula for its determinant, LU factorization and asymptotic spectrum have been studied. Inspired by the results, this paper extends the analysis to the characteristic polynomials and inverse representations. Furthermore, we conduct a comprehensive numerical investigation of asymptotic spectra behavior, validating classical Toeplitz matrix theories through visualization of the spectra and pseudospectra distributions.