Journal articles: 'Kronecker products'

1

Pak, Igor, and Greta Panova. "Unimodality via Kronecker products." Journal of Algebraic Combinatorics 40, no. 4 (April 5, 2014): 1103–20. http://dx.doi.org/10.1007/s10801-014-0520-y.

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2

Zhang, Huamin, and Feng Ding. "On the Kronecker Products and Their Applications." Journal of Applied Mathematics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/296185.

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This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products.

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3

Watts, Valerie L. "Boolean rank of Kronecker products." Linear Algebra and its Applications 336, no. 1-3 (October 2001): 261–64. http://dx.doi.org/10.1016/s0024-3795(01)00338-x.

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4

Brown, Andrew, Stephanie van Willigenburg, and Mike Zabrocki. "Expressions for Catalan Kronecker products." Pacific Journal of Mathematics 248, no. 1 (October 1, 2010): 31–48. http://dx.doi.org/10.2140/pjm.2010.248.31.

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5

Zhao, Hui, Jiuqiang Han, Naiyan Wang, Congfu Xu, and Zhihua Zhang. "A Fast Spectral Relaxation Approach to Matrix Completion via Kronecker Products." Proceedings of the AAAI Conference on Artificial Intelligence 25, no. 1 (August 4, 2011): 580–85. http://dx.doi.org/10.1609/aaai.v25i1.7913.

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In the existing methods for solving matrix completion, such as singular value thresholding (SVT), soft-impute and ﬁxed point continuation (FPCA) algorithms, it is typically required to repeatedly implement singular value decompositions (SVD) of matrices.When the size of the matrix in question is large, the computational complexity of ﬁnding a solution is costly. To reduce this expensive computational complexity, we apply Kronecker products to handle the matrix completion problem. In particular, we propose using Kronecker factorization, which approximates a matrix by the Kronecker product of several matrices of smaller sizes. Weintroduce Kronecker factorization into the soft-impute framework and devise an effective matrix completion algorithm.Especially when the factorized matrices have about the samesizes, the computational complexity of our algorithm is improved substantially.

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6

Pollock, D. Stephen G. "Multidimensional Arrays, Indices and Kronecker Products." Econometrics 9, no. 2 (April 28, 2021): 18. http://dx.doi.org/10.3390/econometrics9020018.

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Much of the algebra that is associated with the Kronecker product of matrices has been rendered in the conventional notation of matrix algebra, which conceals the essential structures of the objects of the analysis. This makes it difficult to establish even the most salient of the results. The problems can be greatly alleviated by adopting an orderly index notation that reveals these structures. This claim is demonstrated by considering a problem that several authors have already addressed without producing a widely accepted solution.

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7

Özel, Mustafa. "Schur Complements of Block Kronecker Products." Deu Muhendislik Fakultesi Fen ve Muhendislik 19, no. 57 (January 1, 2017): 845–50. http://dx.doi.org/10.21205/deufmd.2017195774.

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8

Liu, Jian. "SPECTRAL RADIUS, KRONECKER PRODUCTS AND STATIONARITY." Journal of Time Series Analysis 13, no. 4 (July 1992): 319–25. http://dx.doi.org/10.1111/j.1467-9892.1992.tb00110.x.

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9

Allen, Jeffery C. "Matrix Expansion by Orthogonal Kronecker Products." American Mathematical Monthly 102, no. 6 (June 1995): 538. http://dx.doi.org/10.2307/2974769.

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10

Liechty, Merrill W., and Matthew Tibbits. "Multivariate sufficient statistics using Kronecker products." Statistics and Computing 20, no. 3 (April 23, 2009): 335–41. http://dx.doi.org/10.1007/s11222-009-9127-x.

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11

Allen, Jeffery C. "Matrix Expansion by Orthogonal Kronecker Products." American Mathematical Monthly 102, no. 6 (June 1995): 538–40. http://dx.doi.org/10.1080/00029890.1995.12004614.

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12

Avella-Alaminos, Diana, and Ernesto Vallejo. "Kronecker products and the RSK correspondence." Discrete Mathematics 312, no. 8 (April 2012): 1476–86. http://dx.doi.org/10.1016/j.disc.2012.01.006.

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13

Wang, Wei, and Zhidan Yan. "Connectivity of Kronecker products by K2." Applied Mathematics Letters 25, no. 2 (February 2012): 172–74. http://dx.doi.org/10.1016/j.aml.2011.08.009.

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14

Tian, Yongge, and Ruixia Yuan. "New facts related to dilation factorizations of Kronecker products of matrices." AIMS Mathematics 8, no. 12 (2023): 28818–32. http://dx.doi.org/10.3934/math.20231477.

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<abstract><p>The Kronecker product of two matrices is known as a special algebraic operation of two arbitrary matrices in the computational aspect of matrix theory. This kind of matrix operation has some interesting and striking operation properties, one of which is given by $ (A \otimes B)(C \otimes D) = (AC) \otimes (BD) $ and is often called the mixed-product equality. In view of this equality, the Kronecker product $ A_1 \otimes A_2 $ of any two matrices can be rewritten as the dilation factorization $ A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{n_1} \otimes A_2) $, and the Kronecker product $ A_1 \otimes A_2 \otimes A_3 $ can be rewritten as the dilation factorization $ A_1 \otimes A_2 \otimes A_3 = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{n_1} \otimes A_2 \otimes I_{m_3})(I_{n_1} \otimes I_{n_2} \otimes A_3) $. In this article, we proposed a series of concrete problems regarding the dilation factorizations of the Kronecker products of two or three matrices, and established a collection of novel and pleasing equalities, inequalities, and formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges related to the dilation factorizations. We also present a diverse range of interesting results on the relationships among the Kronecker products $ I_{m_1} \otimes A_2 \otimes A_3 $, $ A_1 \otimes I_{m_2} \otimes A_3 $ and $ A_1 \otimes A_2 \otimes I_{m_3} $.</p></abstract>

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15

Lakshmi N., Vellanki, Jajula Madhu, and Musa Dileep Durani. "On the Kronecker Product of Matrices and Their Applications To Linear Systems Via Modified QR-Algorithm." International Journal of Engineering and Computer Science 10, no. 6 (June 28, 2021): 25352–59. http://dx.doi.org/10.18535/ijecs/v10i6.4600.

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This paper studies and supplements the proofs of the properties of the Kronecker Product of two matrices of different orders. We observe the relation between the singular value decomposition of the matrices and their Kronecker product and the relationship between the determinant, the trace, the rank and the polynomial matrix of the Kronecker products. We also establish the best least square solutions of the Kronecker product system of equations by using modified QR-algorithm.

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16

Pollock, D. S. G. "On Kronecker products, tensor products and matrix differential calculus." International Journal of Computer Mathematics 90, no. 11 (November 2013): 2462–76. http://dx.doi.org/10.1080/00207160.2013.783696.

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17

Gao, Wenlei, Gian Matharu, and Mauricio D. Sacchi. "Fast least-squares reverse time migration via a superposition of Kronecker products." GEOPHYSICS 85, no. 2 (March 1, 2020): S115—S134. http://dx.doi.org/10.1190/geo2019-0254.1.

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Least-squares reverse time migration (LSRTM) has become increasingly popular for complex wavefield imaging due to its ability to equalize image amplitudes, attenuate migration artifacts, handle incomplete and noisy data, and improve spatial resolution. The major drawback of LSRTM is the considerable computational cost incurred by performing migration/demigration at each iteration of the optimization. To ameliorate the computational cost, we introduced a fast method to solve the LSRTM problem in the image domain. Our method is based on a new factorization that approximates the Hessian using a superposition of Kronecker products. The Kronecker factors are small matrices relative to the size of the Hessian. Crucially, the factorization is able to honor the characteristic block-band structure of the Hessian. We have developed a computationally efficient algorithm to estimate the Kronecker factors via low-rank matrix completion. The completion algorithm uses only a small percentage of preferentially sampled elements of the Hessian matrix. Element sampling requires computation of the source and receiver Green’s functions but avoids explicitly constructing the entire Hessian. Our Kronecker-based factorization leads to an imaging technique that we name Kronecker-LSRTM (KLSRTM). The iterative solution of the image-domain KLSRTM is fast because we replace computationally expensive migration/demigration operations with fast matrix multiplications involving small matrices. We first validate the efficacy of our method by explicitly computing the Hessian for a small problem. Subsequent 2D numerical tests compare LSRTM with KLSRTM for several benchmark models. We observe that KLSRTM achieves near-identical images to LSRTM at a significantly reduced computational cost (approximately 5–15× faster); however, KLSRTM has an increased, yet manageable, memory cost.

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18

Tracy, Derrick S., and Kankanam G. Jinadasa. "Partitioned kronecker products of matrices and applications." Canadian Journal of Statistics 17, no. 1 (March 1989): 107–20. http://dx.doi.org/10.2307/3314768.

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19

Hagedorn, George A., and Caroline Lasser. "Symmetric Kronecker Products and Semiclassical Wave Packets." SIAM Journal on Matrix Analysis and Applications 38, no. 4 (January 2017): 1560–79. http://dx.doi.org/10.1137/16m106577x.

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20

Hilberdink, Titus. "Quasi Kronecker products and a determinant formula." Linear Algebra and its Applications 536 (January 2018): 87–102. http://dx.doi.org/10.1016/j.laa.2017.09.011.

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21

Tracy, Derrick S. "Balanced partitioned matrices and their Kronecker products." Computational Statistics & Data Analysis 10, no. 3 (December 1990): 315–23. http://dx.doi.org/10.1016/0167-9473(90)90013-8.

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22

Koning, Ruud H., Heinz Neudecker, and Tom Wansbeek. "Block Kronecker products and the vecb operator." Linear Algebra and its Applications 149 (April 1991): 165–84. http://dx.doi.org/10.1016/0024-3795(91)90332-q.

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23

Castillo, J. E., and R. D. Grone. "Using Kronecker products to construct mimetic gradients." Linear and Multilinear Algebra 65, no. 10 (March 13, 2017): 2031–45. http://dx.doi.org/10.1080/03081087.2017.1298564.

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24

Lin, Wu-Hsiung, and Gerard J. Chang. "Equitable colorings of Kronecker products of graphs." Discrete Applied Mathematics 158, no. 16 (August 2010): 1816–26. http://dx.doi.org/10.1016/j.dam.2010.06.011.

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25

Iosifescu, Mircea, and Horia Scutaru. "Kronecker products, minuscule representations, and polynomial identities." Journal of Mathematical Physics 31, no. 2 (February 1990): 264–77. http://dx.doi.org/10.1063/1.528910.

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26

Fausett, Donald W., and Charles T. Fulton. "Large Least Squares Problems Involving Kronecker Products." SIAM Journal on Matrix Analysis and Applications 15, no. 1 (January 1994): 219–27. http://dx.doi.org/10.1137/s0895479891222106.

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27

Bojanczyk, Adam W., and Adam Lutoborski. "The Procrustes Problem for Orthogonal Kronecker Products." SIAM Journal on Scientific Computing 25, no. 1 (January 2003): 148–63. http://dx.doi.org/10.1137/s1064827501396464.

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28

Ruether, Cameron. "Rost Multipliers of Lifted Kronecker Tensor Products." Documenta Mathematica 25 (2020): 1997–2022. http://dx.doi.org/10.4171/dm/791.

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29

Guo, Litao, Chengfu Qin, and Xiaofeng Guo. "Super connectivity of Kronecker products of graphs." Information Processing Letters 110, no. 16 (July 2010): 659–61. http://dx.doi.org/10.1016/j.ipl.2010.05.013.

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30

Lamping, Frank, Juan-Manuel Peña, and Tomas Sauer. "Spline approximation, Kronecker products and multilinear forms." Numerical Linear Algebra with Applications 23, no. 3 (February 8, 2016): 535–57. http://dx.doi.org/10.1002/nla.2038.

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31

Berele, Allan, and Tom D. Imbo. "Kronecker Products of sn-Characters in Hooks." Journal of Algebra 246, no. 1 (December 2001): 356–66. http://dx.doi.org/10.1006/jabr.2001.8976.

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32

Chen, T. T., and W. Li. "On Condition Numbers for the Weighted Moore-Penrose Inverse and the Weighted Least Squares Problem involving Kronecker Products." East Asian Journal on Applied Mathematics 4, no. 1 (February 2014): 1–20. http://dx.doi.org/10.4208/eajam.230313.070913a.

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AbstractWe establish some explicit expressions for norm-wise, mixed and componentwise condition numbers for the weighted Moore-Penrose inverse of a matrix A ⊗ B and more general matrix function compositions involving Kronecker products. The condition number for the weighted least squares problem (WLS) involving a Kronecker product is also discussed.

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33

Wu, Yi. "A New Method of Kronecker Product Decomposition." Journal of Mathematics 2023 (October 21, 2023): 1–12. http://dx.doi.org/10.1155/2023/9111626.

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Kronecker product decomposition is often applied in various fields such as particle physics, signal processing, image processing, semidefinite programming, quantum computing, and matrix time series analysis. In the paper, a new method of Kronecker product decomposition is proposed. Theoretical results ensure that the new method is convergent and stable. The simulation results show that the new method is far faster than the known method. In fact, the new method is very applicable for exact decomposition, fast decomposition, big matrix decomposition, and online decomposition of Kronecker products. At last, the extension direction of the new method is discussed.

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34

Pad, Pedram, Mohammad Faraji, and Farokh Marvasti. "Constructing and decoding GWBE codes using Kronecker products." IEEE Communications Letters 14, no. 1 (January 2010): 1–3. http://dx.doi.org/10.1109/lcomm.2010.01.091900.

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35

Yun, Yong Sik, and Chul Kang. "SOME RESULTS ON KRONECKER PRODUCTS AND COMMUTATION MATRICES." East Asian mathematical journal 29, no. 3 (June 1, 2013): 259–68. http://dx.doi.org/10.7858/eamj.2013.017.

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36

Gaignaire, R., F. Guyomarc'h, O. Moreau, S. Clenet, and B. Sudret. "Speeding Up SSFEM Computation Using Kronecker Tensor Products." IEEE Transactions on Magnetics 45, no. 3 (March 2009): 1432–35. http://dx.doi.org/10.1109/tmag.2009.2012662.

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37

Dantas, Cassio Fraga, Michele N. da Costa, and Renato da Rocha Lopes. "Learning Dictionaries as a Sum of Kronecker Products." IEEE Signal Processing Letters 24, no. 5 (May 2017): 559–63. http://dx.doi.org/10.1109/lsp.2017.2681159.

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38

Das, Pankaj Kumar, and Lalit K. Vashisht. "Traces of Hadamard and Kronecker products of matrices." Mathematics for Application 6, no. 2 (December 21, 2017): 143–50. http://dx.doi.org/10.13164/ma.2017.09.

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39

Medellín, David, Vivek R. Ravi, and Carlos Torres-Verdín. "Multidimensional NMR inversion without Kronecker products: Multilinear inversion." Journal of Magnetic Resonance 269 (August 2016): 24–35. http://dx.doi.org/10.1016/j.jmr.2016.05.009.

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40

Feng, Jun-e., James Lam, and Yimin Wei. "Spectral properties of sums of certain Kronecker products." Linear Algebra and its Applications 431, no. 9 (October 2009): 1691–701. http://dx.doi.org/10.1016/j.laa.2009.06.004.

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41

Jódar, L., and H. Abou-Kandil. "Kronecker products and coupled matrix Riccati differential systems." Linear Algebra and its Applications 121 (August 1989): 39–51. http://dx.doi.org/10.1016/0024-3795(89)90690-3.

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42

CAO, XIANG-LAN, and ELKIN VUMAR. "SUPER EDGE CONNECTIVITY OF KRONECKER PRODUCTS OF GRAPHS." International Journal of Foundations of Computer Science 25, no. 01 (January 2014): 59–65. http://dx.doi.org/10.1142/s0129054114500038.

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A graph G is said to be super edge connected (in short super – λ) if every minimum edge cut isolates a vertex of G. The Kronecker product of graphs G and H is the graph with vertex set V(G × H) = V(G) × V(H), where two vertices (u1, v1) and (u2, v2) are adjacent in G × H if u1u2 ∈ E(G) and v1v2 ∈ E(H). Let G be a connected graph, and let δ(G) and λ(G) be the minimum degree and the edge-connectivity of G, respectively. In this paper we prove that G × Kn is super-λ for n ≥ 3, if λ(G) = δ(G) and G ≇ K2. Furthermore, we show that K2 × Kn is super-λ when n ≥ 4. Similar results for G × Tn are also obtained, where Tn is the graph obtained from Kn by adding a loop to every vertex of Kn.

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43

Szántó, Csaba. "Submodules of Kronecker modules via extension monoid products." Journal of Pure and Applied Algebra 222, no. 11 (November 2018): 3360–78. http://dx.doi.org/10.1016/j.jpaa.2017.12.012.

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44

Aistleitner, Christoph, Roswitha Hofer, and Gerhard Larcher. "On evil Kronecker sequences and lacunary trigonometric products." Annales de l’institut Fourier 67, no. 2 (2017): 637–87. http://dx.doi.org/10.5802/aif.3094.

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45

Regalia, Phillip A., and Mitra K. Sanjit. "Kronecker Products, Unitary Matrices and Signal Processing Applications." SIAM Review 31, no. 4 (December 1989): 586–613. http://dx.doi.org/10.1137/1031127.

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46

Dirl, R., P. Kasperkovitz, M. I. Aroyo, J. N. Kotzev, and B. L. Davies. "Generating relations for reducing matrices. III. Kronecker products." Journal of Mathematical Physics 28, no. 9 (September 1987): 1947–71. http://dx.doi.org/10.1063/1.527459.

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47

Wang, Wei, and Zhidan Yan. "Connectivity of Kronecker products with complete multipartite graphs." Discrete Applied Mathematics 161, no. 10-11 (July 2013): 1655–59. http://dx.doi.org/10.1016/j.dam.2013.01.009.

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48

Chui, Charles K., and Wenjie He. "Construction of Multivariate Tight Frames via Kronecker Products." Applied and Computational Harmonic Analysis 11, no. 2 (September 2001): 305–12. http://dx.doi.org/10.1006/acha.2001.0355.

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49

Colley, Charles, Huda Nassar, and David F. Gleich. "Dominant Z-Eigenpairs of Tensor Kronecker Products Decouple." SIAM Journal on Matrix Analysis and Applications 44, no. 3 (July 14, 2023): 1006–31. http://dx.doi.org/10.1137/22m1502008.

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50

Gregoria Ariyanti and Ana Easti Rahayu Maya Sari. "The Discrete Lyapunov Equation of The Orthogonal Matrix in Semiring." European Journal of Pure and Applied Mathematics 16, no. 2 (April 30, 2023): 784–90. http://dx.doi.org/10.29020/nybg.ejpam.v16i2.4712.

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Semiring is an algebraic structure of (S, +, ×). Similar to a ring, but without the condition that each element must have an inverse to the adding operation. The forms (S, +) and (S, ×) are semigroups that satisfy the distributive law of multiplication and addition. In matrix theory, there is a term known as the Kronecker product. This operation transforms two matrices into a larger matrix containing all possible products of the entries in the two matrices. This Kronecker product has several properties often used to solve the complex problems of linear algebra and its applications. The Kronecker product is related to the Lyapunov equation of a linear system. Based on previous research in the Lyapunov equation in conventional linear algebra, this paper will describe the characteristics of the Lyapunov equation in a semiring linear system in terms of the Kronecker product.

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Journal articles on the topic 'Kronecker products'

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