The Asymptotic Spectrum of Flipped Multilevel Toeplitz Matrices and of Certain Preconditionings

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

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Toeplitz nonnegative realization of spectra via companion matrices

Special Matrices ◽

10.1515/spma-2019-0017 ◽

2019 ◽

Vol 7 (1) ◽

pp. 230-245

Author(s):

Macarena Collao ◽

Mario Salas ◽

Ricardo L. Soto

Keyword(s):

Eigenvalue Problem ◽

Half Plane ◽

Toeplitz Matrix ◽

Toeplitz Matrices ◽

Nonnegative Matrix ◽

Inverse Eigenvalue Problem ◽

Complex Numbers ◽

Inverse Eigenvalue ◽

Companion Matrices ◽

Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

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Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽

10.3390/sym13050870 ◽

2021 ◽

Vol 13 (5) ◽

pp. 870

Author(s):

Diego Caratelli ◽

Paolo Emilio Ricci

Keyword(s):

Functional Analysis ◽

Computer Algebra ◽

Toeplitz Matrices ◽

Test Cases ◽

Recursive Procedure ◽

Tridiagonal Matrices ◽

Special Cases ◽

The Matrix ◽

Matrix Eigenvalues ◽

Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

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A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

Numerical Algorithms ◽

10.1007/s11075-021-01130-9 ◽

2021 ◽

Author(s):

Sven-Erik Ekström ◽

Paris Vassalos

Keyword(s):

Distribution Function ◽

Asymptotic Distribution ◽

Numerical Experiments ◽

Toeplitz Matrices ◽

Future Research ◽

Research Directions ◽

Numerical Examples ◽

Working Hypothesis ◽

Wide Range ◽

Future Research Directions

AbstractIt is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

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