Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

Wei, Yunlan; Jiang, Xiaoyu; Jiang, Zhaolin; Shon, Sugoog

doi:10.1186/s13662-019-2335-6

Cited by 13 publications

(7 citation statements)

References 32 publications

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“…Another technique which is also different from that used in our work was presented in the paper of Yunlan Wei et al [19], in 2019. In that paper, the authors derived the formulas on representation of the determinants and inverses of the periodic tridiagonal Toeplitz matrices with perturbed corners of type I in the form of products of Fermat numbers and some initial values.…”

Section: Introductionmentioning

confidence: 89%

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Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices

Almeida

Remigio

2023

TCAM

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The characterization of inverses of symmetric tridiagonal and block tridiagonal matrices and the development of algorithms for finding the inverse of any general non-singular tridiagonal matrix are subjects that have been studied by many authors. The results of these research usually depend on the existence of the LU factorization of a non-sigular matrix A, such that A = LU. Besides, the conditions that ensure the nonsingularity of A and its LU factorization are not promptly obtained. Then, we are going to present in this work two extremely simple sufficient conditions for existence of the LU factorization of a Toeplitz symmetric tridiagonal matrix A. We take into consideration the roots of the modified Chebyshev polynomial, and we also present an analysis based on the parameters of the Crout’s method.

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Section: Introductionmentioning

confidence: 89%

“…Recent research continues to highlight the importance of studying Topelitz matrices, such as [3,12,13,19].…”

Section: Introductionmentioning

confidence: 99%

Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices

Almeida

Remigio

2023

TCAM

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“…(21) It is clear to see that the matrix A is also tridiagonal, so its determinant can be calculated by cofactor expansion and be expressed by a recurrence relation [36]. Therefore, in this case, we have…”

Section: Noisy Ou Model With Uniform Samplingmentioning

confidence: 99%

A Framework for Characterising the Value of Information in Hidden Markov Models

Wang¹,

Badiu²,

Coon³

2021

Preprint

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In this paper, a general framework is formalised to characterise the value of information (VoI) in hidden Markov models. Specifically, the VoI is defined as the mutual information between the current, unobserved status at the source and a sequence of observed measurements at the receiver, which can be interpreted as the reduction in the uncertainty of the current status given that we have noisy past observations of a hidden Markov process. We explore the VoI in the context of the noisy Ornstein-Uhlenbeck process and derive its closed-form expressions. Moreover, we study the effect of different sampling policies on VoI, deriving simplified expressions in different noise regimes and analysing the statistical properties of the VoI in the worst case. In simulations, the validity of theoretical results is verified, and the performance of VoI in the Markov and hidden Markov models is also analysed. Numerical results further illustrate that the proposed VoI framework can support the timely transmission in status update systems, and it can also capture the correlation property of the underlying random process and the noise in the transmission environment.

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“…The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices with some special perturbations on the diagonal corners are computed in [9, Section 1.1] and [18]. The determinants and inverses of a family of non-symmetric tridiagonal Toeplitz matrices with perturbed corners are computed in [22].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners

Grudsky¹,

Maximenko²,

Alejandro³

2020

Preprint

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In this paper we study the eigenvalues of Hermitian Toeplitz matrices with the entries 2, −1, 0, . . . , 0, −α in the first column. Notice that the generating symbol depends on the order n of the matrix. If |α| ≤ 1, then the eigenvalues belong to [0, 4] and are asymptotically distributed as the function g(x) = 4 sin 2 (x/2) on [0, π]. The situation changes drastically when |α| > 1 and n tends to infinity. Then the two extreme eigenvalues (the minimal and the maximal one) lay out of [0, 4] and converge rapidly to certain limits determined by the value of α, whilst all others belong to [0, 4] and are asymptotically distributed as g. In all cases, we transform the characteristic equation to a form convenient to solve by numerical methods, and derive asymptotic formulas for the eigenvalues.

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Determinants and inverses of perturbed periodic tridiagonal Toeplitz matrices

Cited by 13 publications

References 32 publications

Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices

Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices

A Framework for Characterising the Value of Information in Hidden Markov Models

Eigenvalues of tridiagonal Hermitian Toeplitz matrices with perturbations in the off-diagonal corners

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