Eigenvalue patterned condition numbers: Toeplitz and Hankel cases

We continue the study started in Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174–189] concerning the sensitivity of simple eigenvalues of a matrix A to perturbations in A   that belong to a chosen subspace of matrices. In Noschese and Pasquini, Eigenvalue condition numbers: zero-structured versus traditional. J. Comput. Appl. Math. 185 (2006) 174–189] the zero-structured perturbations have been considered. Here we focus on patterned perturbations, and the cases of the Toeplitz and of the Hankel matrices are investigated in detail. Useful expressions of the absolute patterned condition number of the eigenvalue λ$\lambda$ and of the analogue of the matrix yx^H${\mathit{yx}}^{\mathrm{H}}$, which leads to the traditional condition number of λ$\lambda$, are given. MATLAB codes are defined to compare traditional, zero-structured and patterned condition numbers. A report on significant numerical tests is included.