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Journal articles on the topic 'Matriz tridiagonal'

Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Rasmawati, Rasmawati, Lailany Yahya, Agusyarif Rezka Nuha, and Resmawan Resmawan. "DETERMINAN SUATU MATRIKS TOEPLITZ K-TRIDIAGONAL MENGGUNAKAN METODE REDUKSI BARIS DAN EKSPANSI KOFAKTOR." Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi 9, no. 1 (April 30, 2021): 6–16. http://dx.doi.org/10.34312/euler.v9i1.10354.

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This paper discusses the determinants of a k-tridiagonal Toeplitz matrix using row reduction and cofactor expansion methods. The analysis was carried out recursively from the general form of the determinant of the tridiagonal Toeplitz matrix, the determinant of the 2-tridiagonal Toeplitz matrix, and the determinant of the 3-tridiagonal Toeplitz matrix. In the end, the general form of the determinant of the k-tridiagonal Toeplitz matrix is obtained.

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2

Kovačec, Alexander. "Schrödinger’s tridiagonal matrix." Special Matrices 9, no. 1 (January 1, 2021): 149–65. http://dx.doi.org/10.1515/spma-2020-0124.

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Abstract In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.

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3

Zgirouski, A. A., and N. A. Likhoded. "Modified method of parallel matrix sweep." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 4 (January 7, 2020): 425–34. http://dx.doi.org/10.29235/1561-2430-2019-55-4-425-434.

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The topic of this paper refers to efficient parallel solvers of block-tridiagonal linear systems of equations. Such systems occur in numerous modeling problems and require usage of high-performance multicore computation systems. One of the widely used methods for solving block-tridiagonal linear systems in parallel is the original block-tridiagonal sweep method. We consider the algorithm based on the partitioning idea. Firstly, the initial matrix is split into parts and transformations are applied to each part independently to obtain equations of a reduced block-tridiagonal system. Secondly, the reduced system is solved sequentially using the classic Thomas algorithm. Finally, all the parts are solved in parallel using the solutions of a reduced system. We propose a modification of this method. It was justified that if known stability conditions for the matrix sweep method are satisfied, then the proposed modification is stable as well.

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4

Fu, Yaru, Xiaoyu Jiang, Zhaolin Jiang, and Seongtae Jhang. "Analytic determinants and inverses of Toeplitz and Hankel tridiagonal matrices with perturbed columns." Special Matrices 8, no. 1 (May 4, 2020): 131–43. http://dx.doi.org/10.1515/spma-2020-0012.

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AbstractIn this paper, our main attention is paid to calculate the determinants and inverses of two types Toeplitz and Hankel tridiagonal matrices with perturbed columns. Specifically, the determinants of the n × n Toeplitz tridiagonal matrices with perturbed columns (type I, II) can be expressed by using the famous Fibonacci numbers, the inverses of Toeplitz tridiagonal matrices with perturbed columns can also be expressed by using the well-known Lucas numbers and four entries in matrix 𝔸. And the determinants of the n×n Hankel tridiagonal matrices with perturbed columns (type I, II) are (−1]) {\left( { - 1} \right)^{{{n\left( {n - 1} \right)} \over 2}}} times of the determinant of the Toeplitz tridiagonal matrix with perturbed columns type I, the entries of the inverses of the Hankel tridiagonal matrices with perturbed columns (type I, II) are the same as that of the inverse of Toeplitz tridiagonal matrix with perturbed columns type I, except the position. In addition, we present some algorithms based on the main theoretical results. Comparison of our new algorithms and some recent works is given. The numerical result confirms our new theoretical results. And we show the superiority of our method by comparing the CPU time of some existing algorithms studied recently.

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5

Dub, P., and O. Litzman. "The Darwin procedure in optics of layered media and the matrix theory." Acta Crystallographica Section A Foundations of Crystallography 55, no. 4 (July 1, 1999): 613–20. http://dx.doi.org/10.1107/s010876739801513x.

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The Darwin dynamical theory of diffraction for two beams yields a nonhomogeneous system of linear algebraic equations with a tridiagonal matrix. It is shown that different formulae of the two-beam Darwin theory can be obtained by a uniform view of the basic properties of tridiagonal matrices, their determinants (continuants) and their close relationship to continued fractions and difference equations. Some remarks concerning the relation of the Darwin theory in the three-beam case to tridiagonal block matrices are also presented.

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6

Pan, Hongyan, and Zhaolin Jiang. "VanderLaan Circulant Type Matrices." Abstract and Applied Analysis 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/329329.

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Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, andg-circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaang-circulant matrix.

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7

Nomura, Kazumasa, and Paul Terwilliger. "Totally bipartite tridiagonal pairs." Electronic Journal of Linear Algebra 37 (June 22, 2021): 434–91. http://dx.doi.org/10.13001/ela.2021.5029.

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There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

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8

Qi, Feng, and Ai-Qi Liu. "Alternative proofs of some formulas for two tridiagonal determinants." Acta Universitatis Sapientiae, Mathematica 10, no. 2 (December 1, 2018): 287–97. http://dx.doi.org/10.2478/ausm-2018-0022.

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Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

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9

Chen, Kwang-Wu. "Horadam Sequences and Tridiagonal Determinants." Symmetry 12, no. 12 (November 28, 2020): 1968. http://dx.doi.org/10.3390/sym12121968.

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We consider a family of particular tridiagonal matrix determinants which can represent the general second-order linear recurrence sequences. These determinants can be changed to symmetric or skew-symmetric tridiagonal determinants. To evaluate the complex factorizations of any Horadam sequence, we evaluate the eigenvalues of some special tridiagonal matrices and their corresponding eigenvectors. We also use these determinant representations to obtain some formulas in these sequences.

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10

Usmani, R. A. "Inversion of Jacobi's tridiagonal matrix." Computers & Mathematics with Applications 27, no. 8 (April 1994): 59–66. http://dx.doi.org/10.1016/0898-1221(94)90066-3.

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11

Takahira, S., T. Sogabe, and T. S. Usuda. "Bidiagonalization of (k, k + 1)-tridiagonal matrices." Special Matrices 7, no. 1 (January 1, 2019): 20–26. http://dx.doi.org/10.1515/spma-2019-0002.

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Abstract In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].

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12

Borowska, Jolanta, and Lena Łacińska. "Eigenvalues of 2-tridiagonal Toeplitz matrix." Journal of Applied Mathematics and Computational Mechanics 14, no. 4 (December 2015): 11–17. http://dx.doi.org/10.17512/jamcm.2015.4.02.

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13

Mayfield, M. Elizabeth. "Perturbation of a Tridiagonal Stability Matrix." Mathematics Magazine 67, no. 2 (April 1, 1994): 124. http://dx.doi.org/10.2307/2690686.

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14

Mallik, Ranjan K. "The inverse of a tridiagonal matrix." Linear Algebra and its Applications 325, no. 1-3 (March 2001): 109–39. http://dx.doi.org/10.1016/s0024-3795(00)00262-7.

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15

Mayfield, M. Elizabeth. "Perturbation of a Tridiagonal Stability Matrix." Mathematics Magazine 67, no. 2 (April 1994): 124–27. http://dx.doi.org/10.1080/0025570x.1994.11996198.

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16

Usmani, Riaz A. "Inversion of a tridiagonal jacobi matrix." Linear Algebra and its Applications 212-213 (November 1994): 413–14. http://dx.doi.org/10.1016/0024-3795(94)90414-6.

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17

Koshcheyeva, I. V., and Yu I. Kuznetsov. "The recreation of a tridiagonal matrix." USSR Computational Mathematics and Mathematical Physics 27, no. 5 (January 1987): 196–99. http://dx.doi.org/10.1016/0041-5553(87)90067-x.

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18

Altun, Muhammed. "Fine Spectra of Symmetric Toeplitz Operators." Abstract and Applied Analysis 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/932785.

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The fine spectra of 2-banded and 3-banded infinite Toeplitz matrices were examined by several authors. The fine spectra ofn-banded triangular Toeplitz matrices and tridiagonal symmetric matrices were computed in the following papers: Altun, “On the fine spectra of triangular toeplitz operators” (2011) and Altun, “Fine spectra of tridiagonal symmetric matrices” (2011). Here, we generalize those results to the ()-banded symmetric Toeplitz matrix operators for arbitrary positive integer .

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19

Yu, Ber-Lin, Zhongshan Li, and Sanzhang Xu. "On the Eventual Exponential Positivity of Some Tree Sign Patterns." Symmetry 13, no. 9 (September 10, 2021): 1669. http://dx.doi.org/10.3390/sym13091669.

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An n×n matrix A is called eventually exponentially positive (EEP) if etA=∑k=0∞tkAkk!>0 for all t≥t0, where t0≥0. A matrix whose entries belong to the set {+,−,0} is called a sign pattern. An n×n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n≤3 are identified. For the n×n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.

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20

Fares, Ali, Ali Ayad, and Bruno de Malafosse. "Calculations on Matrix Transformations Involving an Infinite Tridiagonal Matrix." Axioms 10, no. 3 (September 8, 2021): 218. http://dx.doi.org/10.3390/axioms10030218.

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Given any sequence z=znn≥1 of positive real numbers and any set E of complex sequences, we write Ez for the set of all sequences y=ynn≥1 such that y/z=yn/znn≥1∈E; in particular, sz0 denotes the set of all sequences y such that y/z tends to zero. Here, we consider the infinite tridiagonal matrix Br,s,t˜, obtained from the triangle Br,s,t, by deleting its first row. Then we determine the sets of all positive sequences a=ann≥1 such that EaBr,s,t˜⊂Ea, where E=ℓ∞, c0, or c. These results extend some recent results.

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21

Peleshchak, R. М., V. V. Lytvyn, О. І. Cherniak, І. R. Peleshchak, and М. V. Doroshenko. "STOCHASTIC PSEUDOSPIN NEURAL NETWORK WITH TRIDIAGONAL SYNAPTIC CONNECTIONS." Radio Electronics, Computer Science, Control, no. 2 (July 7, 2021): 114–22. http://dx.doi.org/10.15588/1607-3274-2021-2-12.

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Context. To reduce the computational resource time in the problems of diagnosing and recognizing distorted images based on a fully connected stochastic pseudospin neural network, it becomes necessary to thin out synaptic connections between neurons, which is solved using the method of diagonalizing the matrix of synaptic connections without losing interaction between all neurons in the network. Objective. To create an architecture of a stochastic pseudo-spin neural network with diagonal synaptic connections without loosing the interaction between all the neurons in the layer to reduce its learning time. Method. The paper uses the Hausholder method, the method of compressing input images based on the diagonalization of the matrix of synaptic connections and the computer mathematics system MATLAB for converting a fully connected neural network into a tridiagonal form with hidden synaptic connections between all neurons. Results. We developed a model of a stochastic neural network architecture with sparse renormalized synaptic connections that take into account deleted synaptic connections. Based on the transformation of the synaptic connection matrix of a fully connected neural network into a Hessenberg matrix with tridiagonal synaptic connections, we proposed a renormalized local Hebb rule. Using the computer mathematics system “WolframMathematica 11.3”, we calculated, as a function of the number of neurons N, the relative tuning time of synaptic connections (per iteration) in a stochastic pseudospin neural network with a tridiagonal connection Matrix, relative to the tuning time of synaptic connections (per iteration) in a fully connected synaptic neural network. Conclusions. We found that with an increase in the number of neurons, the tuning time of synaptic connections (per iteration) in a stochastic pseudospin neural network with a tridiagonal connection Matrix, relative to the tuning time of synaptic connections (per iteration) in a fully connected synaptic neural network, decreases according to a hyperbolic law. Depending on the direction of pseudospin neurons, we proposed a classification of a renormalized neural network with a ferromagnetic structure, an antiferromagnetic structure, and a dipole glass.

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22

Da Fonseca, Carlos M. "On a conjecture about a tridiagonal matrix." Journal of Information and Optimization Sciences 41, no. 8 (April 2, 2020): 1789–94. http://dx.doi.org/10.1080/02522667.2019.1696918.

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23

Aneva, Boyka. "Matrix-product ansatz as a tridiagonal algebra." Journal of Physics A: Mathematical and Theoretical 40, no. 39 (September 11, 2007): 11677–95. http://dx.doi.org/10.1088/1751-8113/40/39/001.

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24

Zhang, Deng. "Tridiagonal Random Matrix: Gaussian Fluctuations and Deviations." Journal of Theoretical Probability 30, no. 3 (April 7, 2016): 1076–103. http://dx.doi.org/10.1007/s10959-016-0683-7.

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25

Ran, Rui-sheng, Ting-zhu Huang, Xing-ping Liu, and Tong-xiang Gu. "An inversion algorithm for general tridiagonal matrix." Applied Mathematics and Mechanics 30, no. 2 (February 2009): 247–53. http://dx.doi.org/10.1007/s10483-009-0212-x.

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26

Evans, D. J., and G. M. Megson. "Fast triangularization of a symmetric tridiagonal matrix." Journal of Parallel and Distributed Computing 6, no. 3 (June 1989): 663–78. http://dx.doi.org/10.1016/0743-7315(89)90012-9.

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27

Guseinov, Gusein Sh. "On a Discrete Inverse Problem for Two Spectra." Discrete Dynamics in Nature and Society 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/956407.

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A version of the inverse spectral problem for two spectra of finite-order real Jacobi matrices (tridiagonal symmetric matrices) is investigated. The problem is to reconstruct the matrix using two sets of eigenvalues: one for the original Jacobi matrix and one for the matrix obtained by deleting the last row and last column of the Jacobi matrix.

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28

Zhang, Cai, Bingbing Sun, Jianwei Ma, Huizhu Yang, and Ying Hu. "Splitting algorithms for the high-order compact finite-difference schemes in wave-equation modeling." GEOPHYSICS 81, no. 6 (November 2016): T295—T302. http://dx.doi.org/10.1190/geo2015-0418.1.

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We have developed efficient splitting algorithms for high-order compact finite-difference methods to approximate second-order space derivatives. In general, the methods’ high-order compact finite-difference schemes require the inversion of a multidiagonal matrix that is commonly less efficient. To solve this problem, we used ideas from splitting algorithms in one-way wave-equation migration that work by decomposing the multidiagonal matrix into a series of tridiagonal matrices and then subsequently solving the tridiagonal matrices. This approach results in more efficient algorithms with little loss of accuracy. The splitting algorithms can be implemented in three different ways. Our computational complexity analysis verifies that our methods can reduce the calculation burden from exponential to linear growth. Numerical experiments demonstrate the correctness and effectiveness of our algorithms.

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29

TERWILLIGER, PAUL, and RAIMUNDAS VIDUNAS. "LEONARD PAIRS AND THE ASKEY–WILSON RELATIONS." Journal of Algebra and Its Applications 03, no. 04 (December 2004): 411–26. http://dx.doi.org/10.1142/s0219498804000940.

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Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V which satisfy the following two properties: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Referring to the above Leonard pair, we show there exists a sequence of scalars β,γ,γ*,ϱ,ϱ*,ω,η,η* taken from K such that both [Formula: see text] The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey–Wilson relations.

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30

Benahmed, Boubakeur, Bruno de Malafosse, and Adnan Yassine. "Matrix Transformations and Quasi-Newton Methods." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–17. http://dx.doi.org/10.1155/2007/25704.

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We first recall some properties of infinite tridiagonal matrices considered as matrix transformations in sequence spaces of the formssξ,sξ∘,sξ(c), orlp(ξ). Then, we give some results on the finite section method for approximating a solution of an infinite linear system. Finally, using a quasi-Newton method, we construct a sequence that converges fast to a solution of an infinite linear system.

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31

Da Fonseca, Carlos, Emrah Kılıç, and António Pereira. "The interesting spectral interlacing property for a certain tridiagonal matrix." Electronic Journal of Linear Algebra 36, no. 36 (August 24, 2020): 587–98. http://dx.doi.org/10.13001/ela.2020.4945.

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In this paper, a new tridiagonal matrix, whose eigenvalues are the same as the Sylvester-Kac matrix of the same order, is provided. The interest of this matrix relies also in that the spectrum of a principal submatrix is also of a Sylvester-Kac matrix given rise to an interesting spectral interlacing property. It is proved alternatively that the initial matrix is similar to the Sylvester-Kac matrix.

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32

Chathalingath, Anishchandran, and Arun Manoharan. "Performance Optimization of Tridiagonal Matrix Algorithm [TDMA] on Multicore Architectures." International Journal of Grid and High Performance Computing 11, no. 4 (October 2019): 1–12. http://dx.doi.org/10.4018/ijghpc.2019100101.

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Fast and efficient tridiagonal solvers are highly appreciated in scientific and engineering domain, but challenging optimization task for computer engineers. The state-of-the-art developments in multi-core computing paves the way to meet this challenge to an extent. The technical advances in multi-core computing provide opportunities to exploit lower levels of parallelism and concurrency for inherently sequential algorithms. In this article, the authors present an optimal performance pipelined parallel variant of the conventional Tridiagonal Matrix Algorithm (TDMA), aka the Thomas algorithm, on a multi-core CPU platform. The implementation, analysis and performance comparison of the proposed pipelined parallel TDMA and the conventional version are performed on an Intel SIMD multi-core architecture. The results are compared in terms of elapsed time, speedup, cache miss rate. For a system of ‘n' linear equations where n = 2^36 in presented pipelined parallel TDMA achieves speedup of 1.294X with a parallel efficiency of 43% initially and inclines towards linear speed up as the system grows.

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33

Rojo, Oscar. "The maximal \alpha-index of trees with k pendent vertices and its computation." Electronic Journal of Linear Algebra 36, no. 36 (January 20, 2020): 38–46. http://dx.doi.org/10.13001/ela.2020.5065.

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Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. The $\alpha-$ index of $G$ is the spectral radius $\rho_{\alpha}\left( G\right)$ of the matrix $A_{\alpha}\left( G\right)=\alpha D\left( G\right) +(1-\alpha)A\left( G\right)$ where $\alpha \in [0,1]$. Let $T_{n,k}$ be the tree of order $n$ and $k$ pendent vertices obtained from a star $K_{1,k}$ and $k$ pendent paths of almost equal lengths attached to different pendent vertices of $K_{1,k}$. It is shown that if $\alpha\in\left[ 0,1\right) $ and $T$ is a tree of order $n$ with $k$ pendent vertices then% \[ \rho_{\alpha}(T)\leq\rho_{\alpha}(T_{n,k}), \] with equality holding if and only if $T=T_{n,k}$. This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix ($\alpha=0$). Let $q=[\frac{n-1}{k}]$ and $n-1=kq+r$, $0 \leq r \leq k-1$. It is also obtained that the spectrum of $A_{\alpha}(T_{n,k})$ is the union of the spectra of two special symmetric tridiagonal matrices of order $q$ and $q+1$ when $r=0$ or the union of the spectra of three special symmetric tridiagonal matrices of order $q$, $q+1$ and $2q+2$ when $r \neq 0$. Thus the $\alpha-$ index of $T_{n,k}$ can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order $q+1$ if $r=0$ or order $2q+2$ if $r\neq 0$.

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34

Brimkulov, Ulan. "Matrices whose inverses are tridiagonal, band or block-tridiagonal and their relationship with the covariance matrices of a random Markov process." Filomat 33, no. 5 (2019): 1335–52. http://dx.doi.org/10.2298/fil1905335b.

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The article discusses the matrices of the form A1n, Amn, AmN, whose inverses are: tridiagonal matrix A-1n (n - dimension of the A-mn matrix), banded matrix A-mn (m is the half-width band of the matrix) or block-tridiagonal matrix A-m N (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A1n, Amn, AmN have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.

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35

Igelnik, Boris, and Dan Simon. "The eigenvalues of a tridiagonal matrix in biogeography." Applied Mathematics and Computation 218, no. 1 (September 2011): 195–201. http://dx.doi.org/10.1016/j.amc.2011.05.054.

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36

El-Mikkawy, Moawwad E. A. "On the inverse of a general tridiagonal matrix." Applied Mathematics and Computation 150, no. 3 (March 2004): 669–79. http://dx.doi.org/10.1016/s0096-3003(03)00298-4.

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37

Jiang, Er-Xiong. "Perturbation in eigenvalues of a symmetric tridiagonal matrix." Linear Algebra and its Applications 399 (April 2005): 91–107. http://dx.doi.org/10.1016/j.laa.2004.07.005.

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38

Das, Sandipan Kumar. "Direct solver for pentadiagonal matrix containing tridiagonal submatrices." Numerical Heat Transfer, Part B: Fundamentals 72, no. 1 (July 3, 2017): 1–20. http://dx.doi.org/10.1080/10407790.2017.1338080.

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39

Twig, Y., and R. Kastner. "Block tridiagonal matrix formulation for inhomogeneous penetrable cylinders." IEE Proceedings - Microwaves, Antennas and Propagation 144, no. 3 (1997): 184. http://dx.doi.org/10.1049/ip-map:19971152.

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Sebben, Simone, and B. Rabi Baliga. "SOME EXTENSIONS OF TRIDIAGONAL AND PENTADIAGONAL MATRIX ALGORITHMS." Numerical Heat Transfer, Part B: Fundamentals 28, no. 3 (October 1995): 323–51. http://dx.doi.org/10.1080/10407799508928837.

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Geist, George A. "Reduction of a General Matrix to Tridiagonal Form." SIAM Journal on Matrix Analysis and Applications 12, no. 2 (April 1991): 362–73. http://dx.doi.org/10.1137/0612026.

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Sarayi, Seyyed Mostafa Mousavi Janbeh, Saman Tavana, Morad Karimpour, and Mansour Nikkhah Bahrami. "ANALYTICAL INVERSE FOR THE SYMMETRIC CIRCULANT TRIDIAGONAL MATRIX." Far East Journal of Applied Mathematics 99, no. 1 (May 1, 2018): 1–11. http://dx.doi.org/10.17654/am099010001.

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Pickering, W. M., and M. J. Piff. "On the inverse of a quasi-tridiagonal matrix." International Journal of Computer Mathematics 19, no. 2 (January 1986): 201–6. http://dx.doi.org/10.1080/00207168608803514.

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Sugimoto, Tomoyuki. "On an inverse formula of a tridiagonal matrix." Operators and Matrices, no. 3 (2012): 465–80. http://dx.doi.org/10.7153/oam-06-30.

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Hirshman, S. P., K. S. Perumalla, V. E. Lynch, and R. Sanchez. "BCYCLIC: A parallel block tridiagonal matrix cyclic solver." Journal of Computational Physics 229, no. 18 (September 2010): 6392–404. http://dx.doi.org/10.1016/j.jcp.2010.04.049.

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Chu, Wenchang. "Fibonacci polynomials and Sylvester determinant of tridiagonal matrix." Applied Mathematics and Computation 216, no. 3 (April 2010): 1018–23. http://dx.doi.org/10.1016/j.amc.2010.01.089.

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Feng, Jishe. "Fibonacci identities via the determinant of tridiagonal matrix." Applied Mathematics and Computation 217, no. 12 (February 2011): 5978–81. http://dx.doi.org/10.1016/j.amc.2010.12.025.

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Gündoğar, Zeynep, and Metin Demiralp. "Block tridiagonal matrix enhanced multivariance products representation (BTMEMPR)." Journal of Mathematical Chemistry 56, no. 3 (November 17, 2017): 747–69. http://dx.doi.org/10.1007/s10910-017-0828-7.

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Mattor, Nathan, Timothy J. Williams, and Dennis W. Hewett. "Algorithm for solving tridiagonal matrix problems in parallel." Parallel Computing 21, no. 11 (November 1995): 1769–82. http://dx.doi.org/10.1016/0167-8191(95)00033-0.

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Gover, M. J. C. "The eigenproblem of a tridiagonal 2-Toeplitz matrix." Linear Algebra and its Applications 197-198 (January 1994): 63–78. http://dx.doi.org/10.1016/0024-3795(94)90481-2.

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