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A powerful tool for analyzing and approximating the singular values andeigenvalues of structured matrices is the theory of GLT sequences. By the GLTtheory one can derive a function, which describes the singular value or theeigenvalue distribution of the sequence, the latter under precise assumptions.However, for small values of the matrix size of the considered sequence, theapproximations may not be as good as it is desirable, since in the constructionof the GLT symbol one disregards small norm and low-rank perturbations. On theother hand, LFA can be used to construct polynomial symbols in a similar mannerfor discretizations, where the geometric information is present, but the smallnorm perturbations are retained. The main focus of this paper is theintroduction of the concept of sequence of "Toeplitz momentary symbols",associated with a given sequence of truncated Toeplitz-like matrices. Weconstruct the symbol in the same way as in the GLT theory, but we keep theinformation of the small norm contributions. The low-rank contributions arestill disregarded, and we give an idea on the reason why this is negligible incertain cases and why it is not in other cases, being aware that in presence ofhigh nonnormality the same low-rank perturbation can produce a dramatic changein the eigenvalue distribution. Moreover, a difference with respect to the LFAsymbols is that GLT symbols and Toeplitz momentary symbols are more general andare applicable to a larger class of matrices. We show the applicability of theapproach which leads to higher accuracy in some cases when compared with theGLT symbol. Finally, since for many applications and their analysis it is oftennecessary to consider non-square Toeplitz matrices, we formalize and providesome useful definitions, applicable for non-square Toeplitz momentary symbols.

Toeplitz Momentary Symbols: definition, results, and limitations in the spectral analysis of Structured Matrices