Estimates for the minimum eigenvalue and the condition number of Hermitian (block) Toeplitz matrices

摘要

We give a lower bound for the minimum eigenvalue of the Hermitian Toeplitz matrix ~(Tn)(|θ|~α) and a corresponding upper bound for the spectral condition number ~(κ2)(~(Tn)(|θ|~α)). Our main theorem concerns more general cases and establishes a lower bound for the minimum eigenvalue of ~(Tn)(f) and a corresponding upper bound for ~(κ2)(~(Tn)(f)), provided the non-negative real-valued symbol f satisfies certain conditions. We discuss some examples of symbols for which these estimates work and we see how the minimax principle can be applied together with our main result in order to obtain estimates of λ~(min)(~(Tn)(f)) and ~(κ2)(~(Tn)(f)) even in some cases in which the symbol f does not satisfy the conditions of our main theorem. Finally, we provide an extension of the main result to the block Toeplitz case.