First Degree Rectangular Eigenvalue Problems of Cubic Arrays Over Two Dimensional Ways: A Theoretical Investigation

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1 First Degree Rectangular Eigenvalue Problems of Cubic Arrays Over Two Dimensional Ways: A Theoretical Investigation GİZEM ÖZDEMİR Istanbul Technical University Informatics Institute Ayazağa Campus-İstanbul TÜRKİYE TURKEY) gizmozdemir@gmailcom METİN DEMİRALP Istanbul Technical University Informatics Institute Ayazağa Campus-İstanbul TÜRKİYE TURKEY) metindemiralp@gmailcom Abstract: This work is focused on the rectangular matrices obtained by unfolding multiway arrays The rectangularity depends on how the unfolding procedure is realised In this work we specify the multiway arrays being cubic In other words, we focus on three way arrays and therefore get rectangular matrices whose column numbers is the square of their row numbers The domain of these matrices is spanned by the Kronecker products of vectors whose number of elements match the row number of the considered matrix We focus on the specific eigenvectors whose Kronecker products with a support vector is taken from the domain such that its image under the matrix is proportional to the eigenvector We call these vectors first degree rectangular eigenvectors Key Words: Rectangular eigenvalues and eigenvectors, Multiway arrays, Probabilistic Evolution Theory, Kronecker Product, Unfolding multiway arrays 1 Introduction Probabilistic evolution approach PEA) [1 17] is a recently developed theory for the solution of ODE sets by our group It deals with square and horizontally rectangular matrices The rectangularity is in a way such that the number of columns is a positive integer power of the number of rows If we consider a such matrix then its type can be written as n n j where n is a positive integer while the other positive integer j is greater than 2 A horizontally rectangular matrix of n n j type can be considered as the unfolded [20 23] form of a j + 1) way array each of whose elements can be characterized by j + 1 subindices varying in the same integer set {1, 2,, n} Such multiway arrays can be called Hypercubic arrays because of the domains of their subindices The first way is characterized by the first subindex in such arrays All ways except the first one can be unfolded to a single way, to somehow flatten the hypercubic array onto a plane The result is the horizontally rectangular matrix mentioned above Horizontally rectangular matrices transforms from a Cartesian space whose dimension is the number of the columns in the matrix to another Cartesian space whose dimension matches the row number of the considered matrix Hence, a direct eigenvalue problem in its ordinary meaning can not be defined Instead, singular value problems which are accompanied with two different eigenvalue problems can be defined There has been certain efforts to define the eigenvalues and eigenvectors not by using geometrical considerations but rather through optimisation theory We do not intend to get into the details of these issues here since our purpose is rather geometrical consideration based The paper is organised as follows The second section involves the mathematical background which are rather emphasized on the probabilistic evolution theory, Kronecker power [18,19], folding and unfolding The third section is the core part of the paper It focuses on the first degree rectangular eigenvalue problems The fourth section finalizes the paper with the concluding remarks 2 Mathematical Background 21 Probabilistic Evolution Theory The probabilistic evolution theory involves certain details of probabilistic evolution approach PEA) which converts a given ODE set involving nonlinear structures to a denumerably infinite linear first order homogeneous ODEs with a denumerably infinite square coefficient matrix We can write a single unknown ODE with accompanying appropriate initial conditions ẋt) f xt)) ; x0) a, t [0, ) 1) ISBN:

2 The ODEs with more than one unknowns can also be treated in a similar manner however those cases require the use of multiway arrays In 1) we have assumed that the function f is analytic in a nonempty disk of x complex plane, centered at x x r) The analyticity on the complex plane allows us to use the Taylor series representation which can be expressed as f xt)) j0 f j xt) x r)) j 2) 2) is an infinite linear combination Its summands are linearly independent even though they are functionally dependent So, they can be considered as basis functions We can define x k t) Then we can write xt) x r)) k, k 0, 1, 2, 3) ẋ k t) k k j0 f j xt) x r)) k+j 1 f j x k+j 1 t) 4) j0 Here we have used 1) The matrix form gives an upper Hessenberg form ẋ 0 ẋ 1 ẋ 2 ẋ 3 ẋ f 0 f 1 f 2 f 3 0 2f 0 2f 1 2f f 0 3f f 0 x 0 x 1 x 2 x 3 x 4 5) This matrix 5) is the most general form of the probabilistic evolution approach for a single unknown ODE 22 Kronecker Product The Kronecker product, denoted by, is an operation between two vectorial or matricial or both) entities resulting in a block vector or matrix This outer product is equivalently called Cross Product or Tensor Product It is the result of a special multiplication such as creating blocks by multiplying each element of the first factor with the second one and writing each block at the element s place For a n m type matrix A and a p q type matrix B, the Kronecker product A B is a np mq type block matrix given below a 11 B a 1m B A B a n1 B a nm B a 11 b 11 a 11 b 1q a 1m b 1q a 11 b 21 a 11 b 2q a 1m b 2q a n1 b 11 a n1 b 1q a nm b 1q a n1 b p1 a n1 b pq a nm b pq 6) 23 Folding and Unfolding The folding procedure reorders the elements of a vector one index array) and produces a matrix therefore two index array) For the vector a, if i is even, then the upper half of the vector can be written as the first column of a two column matrix In this case, the second column of this matrix will be formed by the lower half of the vector If the two column matrix is B produced from the vector a, elements of the matrix B can be written with folding procedure as follows, { ai j 1 1 i n/2 b ij 7) a i+n/2 j 2 1 i n/2 The reverse action can be called unfolding when the action starts from the two column matrix to a vector What we have tried to illustrated can be extended to multiway array foldings and unfoldings by performing more rigorous actions 3 First Degree Rectangular Eigenvalue Problems This research focuses on the rectangular matrices that are created by unfolding cubic multiway arrays Here, the dimension of the original multiway array in each direction is assumed to be same In other words, its type is n n n where n is a positive number Hence we focus on the rectangular matrices of n n 2 type where n is a positive integer It is possible to define three different eigenvalue problems for such a matrix A in probabilistic evolution approach theory One may consider those as of first and second degree First degree eigenvalue problems can be defined ISBN:

3 as Au a) αa Aa u) αa 8) where α denotes the eigenvalue while a stands for the eigenvector having n elements, that is, a matrix with the form n 1, and u is a support vector with n elements represents the Kronecker product Second degree eigenvalue problems can be defined as follows, Aa a) Aa 2 αa 9) In this work, we will deal with the first degree eigenvalue problems By definition we have to consider each of those equations as different problems Each of them produces n homogenous equations that contains n unknowns In fact, these equations construct the kernel space of a square coefficient matrix It is required that this kernel space is not empty to be able to find a nonvanishing a αs that satisfy this property will be the eigenvalues The eigenvalue problem will be solved by finding both the αs and the corresponding a vector This research will examine the case where n 2 A a 111 a 112 a 211 a 212 a 121 a 122 a 221 a 222 We can reindexing matrix A A11 A A u 12 A 13 A 14 A 21 A 22 A 23 A 24 10) 11) When we apply the Kronecker product to support vector u and eigenvector a, the following equation is obtained u a [ u1 a u 2 a ] Then, we multiply A u with u a u 1 a 1 u 1 a 2 u 2 a 1 u 2 a 2 12) A u u a) A11 u 1 a 1 + A 12 u 1 a 2 + A 13 u 2 a 1 + A 14 u 2 a 2 A 21 u 1 a 1 + A 22 u 1 a 2 + A 23 u 2 a 1 + A 24 u 2 a 2 A11 u 1 + A 13 u 2 A 12 u 1 + A 14 u 2 a1 A 22 u 1 + A 24 u 2 a 2 13) From the equation of the first order eigenvalue problems, 13) will be equal to αa Now, let A11 u A w 1 + A 13 u 2 A 12 u 1 + A 14 u 2 14) A 22 u 1 + A 24 u 2 Therefore, A w a αa Now, by using this statement, eigenvalues and eigenvectors are desired to determine Then we denote as, A 11 u 1 A 13 u 2 A 22 u 1 A 24 u 2 ) 2 4 A 11 A 22 u A 11 A 24 u 1 u 2 +A 13 A 22 u 1 u 2 + A 13 A 24 u 2 2 A 12 A 21 u 2 1 A 12 A 23 u 1 u 2 A 14 A 21 u 1 u 2 A 14 A 23 u 2 ) 2 15) Thus, we can find the eigenvalues from the characteristic polynomial α A 11u 1 + A 13 u 2 + A 22 u 1 + A 24 u 2 ) α A 11u 1 + A 13 u 2 + A 22 u 1 + A 24 u 2 + ) 16) We obtain eigenvectors with using eigenvalues When we take a 2 1, then a 1 A 22u 1 + A 24 u 2 α 17) Here, for α 1 a 1 A 22u 1 + A 24 u 2 α 1 18) W α1 {a 1, 1) C 2 } { A )} 22u 1 + A 24 u 2 α 1, 1 19) With a similar way, for α 2 W α2 {a 1, 1) C 2 } { A )} 22u 1 + A 24 u 2 α 2, 1 20) 4 Conclusion This work focused on the first degree rectangular eigenvalues of unfolded cubic multiway arrays We have defined two different versions of this problems What we have obtained is that these two different rectangular eigenvalue problems may not have matching eigenvectors and even eigenvalues unless the original multiway array has certain specific symmetries We have not intended to proceed to further details of this issue here since this work have aimed to give mother lines of the eigenpair concept Future works from our groups will focus on further important findings and details ISBN:

4 References: [1] F Hunutlu, N A Baykara and M Demiralp Truncation Approximants to Probabilistic Evolution of ODEs Having Two Diagonal Banded Evolution Matrices Under Initial Conditions: Simple Case, Twelfth International Conference on Computational and Mathematical Methods in Sciences and Engineering La Manga, Spain, 2012) [2] M Demiralp and E Demiralp Probabilistic evolutions: The ultimate, natural and exact linearisation of ODEs In AIP Proceedings for the 9th International Conference on Computational Methods in Science and Engineering IC- CMSE2011), page in print, Halkidiki, Greece, 2-7 October 2011 [3] M Demiralp and E Demiralp Increasing the qualities of the truncated probabilistic evolutions by controlling fluctuations In AIP Proceedings for the 9th International Conference on Computational Methods in Science and Engineering ICCMSE2011), page in print, Halkidiki, Greece, 2-7 October 2011 [4] E Gürvit, N A Baykara, and M Demiralp Employing the probabilistic evolutions of ODEs in function inversion under fluctuation control In AIP Proceedings for the 9th International Conference on Computational Methods in Science and Engineering ICCMSE2011), page in print, Halkidiki, Greece, 2-7 October 2011 [5] S Bayat and M Demiralp Probabilistic evolution for the most general first order single unknown explicit ODEs: Autonomization, triangularization, and, certain important aspects in the analysis In R Raducanu, N Mastorakis, R Neck, V Niola, and Ka-Lok Ng, editors, Proceedings and Chemistry MCBC 12), ISBN: , pages 5762, Iasi, Romania, June 2012 WSEAS Press [6] N A Baykara and M Demiralp Taking care of the singularities in the probabilistic evolutionary quantum expectation value dynamics In J Vigo- Aguiar, editor, Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012, volume 1 of ISBN: , pages , Murcia, Spain, 2-5 July 2012 [7] D Bodur and M Demiralp Probabilistic evolution approach to first order explicit ordinary differential equations for two unknown case In D Biolek and N A Baykara, editors, Proceedings of the 12th WSEAS International Conference on Systems Theory and Scientific Computation ISTASC12), ISBN: , pages , Istanbul, Turkey, August 2012 WSEAS Press [8] M Demiralp Solving initial value problems of multivariable parabolic systems via expectation values: Probabilistic evolution, exactness and approximants In R Raducanu, N Mastorakis, R Neck, V Niola, and Ka-Lok Ng, editors, Proceedings and Chemistry MCBC 12), ISBN: , pages 1415, Iasi, Romania, June 2012 WSEAS Press Keynote Speech [9] M Demiralp Quantum expected value dynamics in probabilistic evolution perspective In J Vigo-Aguiar, editor, Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012, volume 2 of ISBN: , pages , Murcia, Spain, 2-5 July 2012 [10] C Gözükırmızı and M Demiralp Convergence of probabilistic evolution truncation approximants via eigenfunctions of evolution operator In R Raducanu, N Mastorakis, R Neck, V Niola, and Ka-Lok Ng, editors, Proceedings and Chemistry MCBC 12), ISBN: , pages 4550, Iasi, Romania, June 2012 WSEAS Press [11] C Gözükırmızı and M Demiralp Analytic continuation possibilities for divergent initial vectors in the probabilistic evolution of explicit ODEs In D Biolek and N A Baykara, editors, Proceedings of the 12th WSEAS International Computation ISTASC12), ISBN: , pages , Istanbul, Turkey, August 2012 WSEAS Press [12] E Gürvit and M Demiralp Enhanced multivariate product representation at constancy level in probabilistic evolution approach to first order explicit ODEs In D Biolek and N A Baykara, ISBN:

5 editors, Proceedings of the 12th WSEAS International Computation ISTASC12), ISBN: , pages , Istanbul, Turkey, August 2012 WSEAS Press [13] F Hunutlu, N A Baykara, and M Demiralp Conicalization of the probabilistic evolutions for the ordinary and forced Van Der Pol equation under given initial conditions In R Raducanu, N Mastorakis, R Neck, V Niola, and Ka-Lok Ng, editors, Proceedings of the 13th WSEAS International Conference on Mathematics and Computers in Biology and Chemistry MCBC 12), ISBN: , pages 3944, Iasi, Romania, June 2012 WSEAS Press [14] F Hunutlu, N A Baykara, and M Demiralp Truncation approximants to probabilistic evolution for ODEs having two diagonal banded evolution matrices under initial conditions: Simple case In J Vigo- Aguiar, editor, Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012, volume 2 of ISBN: , pages , Murcia, Spain, 2-5 July 2012 [15] F Hunutlu, N A Baykara, and M Demiralp Truncation approximants and their qualities in the probabilistic evolution of Van Der Pol equation under initial conditions In D Biolek and N A Baykara, editors, Proceedings of the 12th WSEAS International Conference on Systems Theory and Scientific Computation ISTASC12), ISBN: , pages , Istanbul, Turkey, August 2012 WSEAS Press [16] B Kalay and M Demiralp Quantum expected value dynamics in probabilistic evolution perspective for systems under dynamic weak external fields In D Biolek and N A Baykara, editors, Proceedings of the 12th WSEAS International Computation ISTASC12), ISBN: , pages , Istanbul, Turkey, August 2012 WSEAS Press [17] S Tuna and M Demiralp Certain validations of probabilistic evolution approach for initial value problems In D Biolek and N A Baykara, editors, Proceedings of the 12th WSEAS International Conference on Systems Theory and Scientific Computation ISTASC12), ISBN: , pages , Istanbul, Turkey, August 2012 WSEAS Press [18] D Bodur, M Demiralp, Probabilistic Evolution Approach to First Order Explicit Ordinary Differential Equations for Two Unknown Case, WSEAS International Conference on Applied Mathematics Istanbul, Turkey, 2012) [19] Charles FVan Loan The ubiquitous Kronecker product Journal of Computational and Applied Mathematics Volume 123, Issues 12, 1 November 2000, Pages [20] S Tuna, N A Baykara, and M Demiralp Weighted singular value decomposition for folded matrices In N Mastorakis, M Demiralp, and N A Baykara, editors, Proceedings of the 2nd International Conference on Applied Informatics and Computing Theory AICT11), IEEEAM, ISBN: , pages 7075, Prague, Czech Republic, September 2011 [21] M Demiralp Tensors or folvecs, folmats and folarrs: Welcome to the enchanted realm of linearity In Proceedings of the International Conference on Environment, Economics, Energy, Devices, Systems, Communications, Computers, Mathematics, ISBN: , pages 1010, Drobeta Turnu Severin, Romania, October 2011 Keynote Speech [22] L Divanyan and M Demiralp High dimensional model representation hdmr) based folded vector decomposition In N Mastorakis, M Demiralp, and N A Baykara, editors, Proceedings of the 2nd International Conference on Applied Informatics and Computing Theory AICT11), IEEEAM, ISBN: , pages 3944, Prague, Czech Republic, September 2011 [23] M Demiralp Folding and unfolding related issues, especially decompositions, in data processing In S Sendra and J C Metrolho, editors, Proceedings of the 16th WSEAS International Conference on Computers part of CSCC12), ISBN: , Kos, Greece, July 2012 Keynote Speech ISBN: