34 research outputs found

    Special least squares solutions of the reduced biquaternion matrix equation with applications

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    This paper presents an efficient method for obtaining the least squares Hermitian solutions of the reduced biquaternion matrix equation (AXB,CXD)=(E,F)(AXB, CXD) = (E, F ). The method leverages the real representation of reduced biquaternion matrices. Furthermore, we establish the necessary and sufficient conditions for the existence and uniqueness of the Hermitian solution, along with a general expression for it. Notably, this approach differs from the one previously developed by Yuan et al. (2020)(2020), which relied on the complex representation of reduced biquaternion matrices. In contrast, our method exclusively employs real matrices and utilizes real arithmetic operations, resulting in enhanced efficiency. We also apply our developed framework to find the Hermitian solutions for the complex matrix equation (AXB,CXD)=(E,F)(AXB, CXD) = (E, F ), expanding its utility in addressing inverse problems. Specifically, we investigate its effectiveness in addressing partially described inverse eigenvalue problems. Finally, we provide numerical examples to demonstrate the effectiveness of our method and its superiority over the existing approach.Comment: 25 pages, 3 figure

    L-structure least squares solutions of reduced biquaternion matrix equations with applications

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    This paper presents a framework for computing the structure-constrained least squares solutions to the generalized reduced biquaternion matrix equations (RBMEs). The investigation focuses on three different matrix equations: a linear matrix equation with multiple unknown L-structures, a linear matrix equation with one unknown L-structure, and the general coupled linear matrix equations with one unknown L-structure. Our approach leverages the complex representation of reduced biquaternion matrices. To showcase the versatility of the developed framework, we utilize it to find structure-constrained solutions for complex and real matrix equations, broadening its applicability to various inverse problems. Specifically, we explore its utility in addressing partially described inverse eigenvalue problems (PDIEPs) and generalized PDIEPs. Our study concludes with numerical examples.Comment: 30 page

    Matris denklemleri ile ilişkili bazı özel tipli matrisler için matris yakınlık problemi

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    06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Anahtar Kelimeler: minimum kalan problemi, matris yakınlık problemi, en iyi yaklaşık çözüm, Moore-Penrose ters. İlk bölümde lineer matris denklem problemleri ile ilgili literatür bilgisine yer verilmiş ve çalışmanın içeriğini oluşturan problemler tanıtılmıştır. İkinci bölümde çalışmada kullanılan bazı tanımlar ve temel teoremlerden bahsedilmiştir. Üçüncü bölümün ilk kısmında (A1XB1,A2XB2,,AkXBk)=(C1,C2,,Ck)\left( {{A_1}X{B_1},{A_2}X{B_2}, \ldots ,{A_k}X{B_k}} \right) = \left( {{C_1},{C_2}, \ldots ,{C_k}} \right) matris denkleminin simetrik ve ters-simetrik matrisler için genel çözümlerinin kümesi ve en küçük kareler çözümlerinin kümesi, Moore-Penrose ters ve Kronecker çarpım kullanılarak incelenmiştir. Bu matris denkleminin en iyi yaklaşık simetrik çözümü ve en iyi yaklaşık ters-simetrik çözümü ortaya konulmuştur. İkinci kısmında AXB=C matris denkleminin (P,Q)-ortogonal simetrik ve (P,Q)-ortogonal ters-simetrik matrisler için genel çözümlerinin kümesi ve en küçük kareler çözümlerinin kümesi Moore-Penrose ters ve spektral ayrışım kullanılarak incelenmiştir. Daha sonra, en iyi yaklaşık (P,Q)-ortogonal simetrik çözümü ve (P,Q)-ortogonal ters-simetrik çözümü elde edilmiştir. Son olarak, her iki kısmın sonunda ele alınan problemlerin çözümünü elde etmek için kullanılan bir algoritma, iki örnek ve literatürden seçilmiş örnekler için karşılaştırmalı bir tablo verilmiştir. Dördüncü bölümde (AXB,CXD)=(E,F) kuaterniyon matris denkleminin merkezi-hermityen ve ters-merkezi-hermityen matrisler üzerinde minimum kalan problemi Moore-Penrose ters, Kronecker çarpım ve vec operatörü kullanılarak incelenmiştir. Daha sonra ise (AXB,CXD)=(E,F) kuaterniyon matris denkleminin en iyi yaklaşık merkezi-hermityen çözümü ve ters-merkezi-hermityen çözümü verilmiştir. Son olarak, bölüm sonunda ele alınan problemlerin çözümünü elde etmek için kullanılan bir algoritma ve iki sayısal örnek verilmiştir. Son bölüm ise sonuçların kısa bir tartışmasına ayrılmıştır.Yeni kaotik sistemin FPGA tabalı tasarım

    Algebraic technique for mixed least squares and total least squares problem in the reduced biquaternion algebra

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    This paper presents the reduced biquaternion mixed least squares and total least squares (RBMTLS) method for solving an overdetermined system AXBAX \approx B in the reduced biquaternion algebra. The RBMTLS method is suitable when matrix BB and a few columns of matrix AA contain errors. By examining real representations of reduced biquaternion matrices, we investigate the conditions for the existence and uniqueness of the real RBMTLS solution and derive an explicit expression for the real RBMTLS solution. The proposed technique covers two special cases: the reduced biquaternion total least squares (RBTLS) method and the reduced biquaternion least squares (RBLS) method. Furthermore, the developed method is also used to find the best approximate solution to AXBAX \approx B over a complex field. Lastly, a numerical example is presented to support our findings.Comment: 19 pages, 3 figure

    Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

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    Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer’s rule) of a partial solution to the system of two-sided quaternion matrix equations A1XB1=C1, A2XB2=C2. We also give Cramer’s rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A1X=C1 and XB1=C1, respectively, and with an unchanging second equation. Cramer’s rules for special cases when two equations are one-sided, namely the system of the equations A1X=C1, XB2=C2, and the system of the equations A1X=C1, A2X=C2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well

    Toward Solution of Matrix Equation X=Af(X)B+C

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    This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation X=Af(X)B+CX=Af(X) B+C with f(X)=XT,f(X) =X^{\mathrm{T}}, f(X)=Xˉf(X) =\bar{X} and f(X)=XH,f(X) =X^{\mathrm{H}}, where XX is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of W=AWB+CW=\mathcal{A}W\mathcal{B}+\mathcal{C} where the dimensions of the coefficient matrices A,B\mathcal{A},\mathcal{B} and C\mathcal{C} are the same as those of the original equation. Closed-form solutions of equation X=Af(X)B+CX=Af(X) B+C can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation equation X=Af(X)B+CX=Af(X) B+C. Necessary and sufficient conditions are established to guarantee the convergence of the iterations

    Solving and Algorithm for Least-Norm General Solution to Constrained Sylvester Matrix Equation

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    Keeping in view that a lot of physical systems with inverse problems can be written by matrix equations, the least-norm of the solution to a general Sylvester matrix equation with restrictions A1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4,A3X1B3+A4X2B4=Cc, is researched in this chapter. A novel expression of the general solution to this system is established and necessary and sufficient conditions for its existence are constituted. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore–Penrose inverse previously obtained by one of the authors

    Mixed Witt rings of algebras with involution of the first kind

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    The Witt ring W(K)W(K) of a field is a central object in the algebraic theory of quadratic forms. In general, we can define a Witt ring for commutative rings with involution. But if (A,σ)(A,\sigma) is a central simple algebra with involution of the first over a field KK, we only have Witt groups Wε(A,σ)W^\varepsilon(A,\sigma), mainly because AA is not commutative, so there is no appropriate tensor product of modules. We use the fact that there is a hermitian Morita equivalence between (AKA,σσ)(A\otimes_K A,\sigma\otimes \sigma) and (K,Id)(K,Id) to define a commutative graded ring structure on W~(A,σ)=W(K)W+(A,σ)W(A,σ)\widetilde{W}(A,\sigma) = W(K) \oplus W^+(A,\sigma)\oplus W^-(A,\sigma), the main difficulty being the associativity. We study the basic properties of this ring (the mixed Witt ring of (A,σ)(A,\sigma)), as well as the mixed Grothendieck-Witt ring GW~(A,σ)\widetilde{GW}(A,\sigma). In particular, GW~(A,σ)\widetilde{GW}(A,\sigma) is a pre-λ\lambda-ring for some exterior power operations defined through hermitian Morita equivalences. The mixed Witt ring has a fundamental filtration, and the associated graded ring has strong links with modulo 2 Galois cohomology. We also classify the prime ideals of W~(A,σ)\widetilde{W}(A,\sigma) and relate this to the previous studies on signatures for hermitian forms
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