Advances in Researches of Quaternion Algebras
Open Access
Review
Issue
4open
Volume 2, 2019
Advances in Researches of Quaternion Algebras
Article Number 24
Number of page(s) 15
Section Mathematics - Applied Mathematics
DOI https://doi.org/10.1051/fopen/2019021
Published online 09 July 2019
  • Handson A, Hui H (1995), Quaternion frame approach to streamline visualization. IEEE Trans Vis Computer Graph 1, 2, 164–172. [CrossRef] [Google Scholar]
  • Sangwine SJ (1996), Fourier transforms of colour images using quaternion or hypercomplex number. Electr Lett 32, 21, 1979–1980. [CrossRef] [Google Scholar]
  • Ell TA, Sangwine SJ (2007), Hypercomplex Fourier transforms of color images. IEEE Trans Image Process 16, 1, 22–35. [CrossRef] [PubMed] [Google Scholar]
  • Wang J, Li T, Shi YQ, Lian S, Ye J (2016), Forensics feature analysis in quaternion wavelet domain for distinguishing photographic images and computer graphics. Multimedia Tools Appl 76, 22, 23712–23737. [Google Scholar]
  • Chen B, Shu H, Coatrieux G, Chen G, Sun X, Coatrieux JL (2015), Color image analysis by quaternion-type moments. J Math Imaging Vision 51, 1, 124–144. [CrossRef] [Google Scholar]
  • Gibbon JD, Holm DD, Kerr RM, Roulstone I (2006), Quaternions and particle dynamics in the Euler fluid equations. Nonlinearity 19, 1969–1983. [Google Scholar]
  • Roubtsov VN, Roulstone I (2001), Holomorphic structures in hydrodynamical models of nearly geostrophic flow. Proc R Society London Ser A 457, 1519–1531. [CrossRef] [Google Scholar]
  • Adler SL (1995), Quaternionic quantum mechanics and quantum fields, Oxford University Press, New York. [Google Scholar]
  • Leo S, Ducati G (2012), Delay time in quaternionic quantum mechanics. J Math Phys 53, 2, Article ID 022102, 8 p. [PubMed] [Google Scholar]
  • Gupta S (1998), Linear quaternion equations with application to spacecraft attitude propagation. IEEE Aerospace Conf Proc 1, 69–76. [Google Scholar]
  • Song C, Sang J, Seung H, Nam HS (2006), Robust control of the missile attitude based on quaternion feedback. Control Eng Pract 14, 7, 811–818. [Google Scholar]
  • Wie B, Weiss H, Arapostathis A (1989), Quaternion feedback regulator for spacecraft eigenaxis rotations. J Guidance Control Dyn 12, 375–380. [CrossRef] [Google Scholar]
  • Took CC, Mandic DP (2009), The quaternion LMS algorithm for adaptive filtering of hypercomplex real world processes. IEEE Trans Signal Process 57, 4, 1316–1327. [Google Scholar]
  • Took CC, Mandic DP (2010), A quaternion widely linear adaptive filter. IEEE Trans Signal Process 58, 8, 4427–4431. [Google Scholar]
  • Took CC, Mandic DP (2011), Augmented second-order statistics of quaternion random signals. Signal Process 91, 214–224. [CrossRef] [Google Scholar]
  • Mitra SK (1973), A pair of simultaneous linear matrix A1XB1 = C1 and A2XB2 = C2. Proc Camb Philos Soc 74, 213–216. [Google Scholar]
  • Mitra SK (1990), A pair of simultaneous linear matrix equations and a matrix programming problem. Linear Algebra Its Appl 131, 97–123. [CrossRef] [Google Scholar]
  • Shinozaki N, Sibuya M (1974), Consistency of a pair of matrix equations with an application. Keio Eng Rep 27, 141–146. [Google Scholar]
  • Özgüler AB, Akar N (1991), A common solution to a pair of linear matrix equations over a principal domain. Linear Algebra Its Appl 144, 85–99. [CrossRef] [Google Scholar]
  • Navarra A, Odell PL, Young DM (2001), A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications. Comput Math Appl 41, 929–935. [Google Scholar]
  • Wang QW (2005), The general solution to a system of real quaternion matrix equations. Comput Math Appl 49, 665–675. [Google Scholar]
  • Horn RA, Zhang F (2012), A generalization of the complex Autonne-Takagi factorization to quaternion matrices. Linear Multilinear Algebra 60, 11–12, 1239–1244. [CrossRef] [Google Scholar]
  • Took CC, Mandic DP, Zhang FZ (2011), On the unitary diagonalization of a special class of quaternion matrices. Appl Math Lett 24, 1806–1809. [Google Scholar]
  • Yuan SF, Wang QW (2012), Two special kinds of least squares solutions for the quaternion matrix equation AXB + CXD = E. Electr J Linear Algebra 23, 257–274. [Google Scholar]
  • Liu X (2018), The η-anti-Hermitian solution to some classic matrix equations. Appl Math Comput 320, 264–270. [Google Scholar]
  • He ZH, Wang QW (2013), A real quaternion matrix equation with applications. Linear Multilinear Algebra 61, 6, 725–740. [CrossRef] [Google Scholar]
  • Beik FPA, Ahmadi-As S (2015), An iterative algorithm for η-(anti)-Hermitian least-squares solutions of quaternion matrix equations. Electr J Linear Algebra 30, 372–401. [Google Scholar]
  • Futorny V, Klymchuk T, Sergeichuk VV (2016), Roth’s solvability criteria for the matrix equations AX-Formula B = C and X-AFormula B = C over the skew field of quaternions with an involutive automorphism qFormula . Linear Algebra Appl 510, 246–258. [Google Scholar]
  • He ZH, Wang QW (2014), The η-bihermitian solution to a system of real quaternion matrix equations. Linear Multilinear Algebra 62, 11, 1509–1528. [CrossRef] [Google Scholar]
  • He ZH, Wang QW, Zhang Y (2017), Simultaneous decomposition of quaternion matrices involving η-Hermicity with applications. Applied Math Comput 298, 13–35. [Google Scholar]
  • He ZH, Liu J, Tam TY (2017), The general ϕ-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations. Electr J Linear Algebra 32, 475–499. [CrossRef] [Google Scholar]
  • He ZH (2019), Structure, properties and applications of some simultaneous decompositions for quaternion matrices involving ϕ-skew-Hermicity. Adv Appl Clifford Algebras 29, 6. [CrossRef] [Google Scholar]
  • Klimchuk T, Sergeichuk VV (2014), Consimilarity and quaternion matrix equations AX-Formula B = C and X-AFormula B = C. Special Matrices 2, 180–186. [CrossRef] [Google Scholar]
  • Rodman L (2014), Topics in Quaternion Linear Algebra, Princeton University Press, Princeton. [Google Scholar]
  • Rehman A, Wang QW, He ZH (2015), Solution to a system of real quaternion matrix equations encompassing η-Hermicity. Appl Math Comput 265, 945–957. [Google Scholar]
  • Rehman A, Wang QW, Ali I, Akram M, Ahmad MO (2017), A constraint system of generalized Sylvester quaternion matrix equations. Adv Appl Clifford Algebras 27, 4, 3183–3196. [CrossRef] [Google Scholar]
  • Zhang Y, Wang RH (2013), The exact solution of a system of quaternion matrix equations involving η-Hermicity. Appl Math Comput 222, 201–209. [Google Scholar]
  • Yuan SF, Wang QW, Xiong ZP (2014), The least squares eta-Hermitian problems of quaternion matrix equation. Filomat 28, 6, 1153–1165. [CrossRef] [Google Scholar]
  • Bapat RB, Bhaskara KPS, Prasad KM (1990), Generalized inverses over integral domains. Linear Algebra Appl 140, 181–196. [Google Scholar]
  • Stanimirovic PS (1996), General determinantal representation of pseudoinverses of matrices. Matematički Vesnik 48, 1–9. [Google Scholar]
  • Kyrchei I (2015), Cramer’s rule for generalized inverse solutions, in: I Kyrchei (Ed.), Advances in Linear Algebra Research, Nova Science Publishers, New York, pp. 79–132. [Google Scholar]
  • Kyrchei I (2008), Cramer’s rule for quaternion systems of linear equations. J Math Sci 155, 6, 839–858. [CrossRef] [Google Scholar]
  • Kyrchei I (2012), The theory of the column and row determinants in a quaternion linear algebra, in: AR Baswell (Ed.), Advances in Mathematics Research, 15, Nova Science Publishers, New York, pp. 301–359. [Google Scholar]
  • Kyrchei I (2011), Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer’s rules. Linear Multilinear Algebra 59, 4, 413–431. [CrossRef] [Google Scholar]
  • Kyrchei I (2013), Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations. Linear Algebra Appl 438, 1, 136–152. [Google Scholar]
  • Kyrchei I (2014), Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Appl Math Comput 238, 193–207. [Google Scholar]
  • Kyrchei I (2015), Determinantal representations of the W-weighted Drazin inverse over the quaternion skew field. Appl Math Comput 264, 453–465. [Google Scholar]
  • Kyrchei I (2016), Explicit determinantal representation formulas of W-weighted Drazin inverse solutions of some matrix equations over the quaternion skew field. Math Probl Eng 2016, Article ID 8673809, 13 p. [Google Scholar]
  • Kyrchei I (2017), Determinantal representations of the Drazin and W-weighted Drazin inverses over the quaternion skew field with applications, in: S. Griffin (Ed.), Quaternions: Theory and Applications, Nova Science Publishers, New York, pp. 201–275. [Google Scholar]
  • Kyrchei I (2017), Weighted singular value decomposition and determinantal representations of the quaternion weighted Moore-Penrose inverse. Appl Math Comput 309, 1–16. [Google Scholar]
  • Kyrchei I (2017), Determinantal representations of the quaternion weighted Moore-Penrose inverse and its applications, in: AR Baswell (Ed.), Advances in Mathematics Research, 23, Nova Science Publications, New York, pp. 35–96. [Google Scholar]
  • Kyrchei I (2018), Explicit determinantal representation formulas for the solution of the two-sided restricted quaternionic matrix equation. J Appl Math Comput 58, 1–2, 335–365. [Google Scholar]
  • Kyrchei I (2018), Determinantal representations of solutions to systems of quaternion matrix equations. Adv Appl Clifford Algebras 28, 1, 23. [CrossRef] [Google Scholar]
  • Kyrchei I (2018), Cramer’s rules for Sylvester quaternion matrix equation and its special cases. Adv Appl Clifford Algebras 28, 5, 90. [CrossRef] [Google Scholar]
  • Kyrchei I (2018), Determinantal representations of solutions and Hermitian solutions to some system of two-sided quaternion matrix equations. J Math 2018, ID 6294672, 12 p. [CrossRef] [Google Scholar]
  • Kyrchei I (2018), Cramer’s rules for the system of two-sided matrix equations and of its special cases, in: HA Yasser (Ed.), Matrix Theory-Applications and Theorems, IntechOpen, London, pp. 3–20. [Google Scholar]
  • Kyrchei I (2019), Determinantal representations of general and (skew-)Hermitian solutions to the generalized Sylvester-type quaternion matrix equation. Abstract Appl Anal 2019, Article ID 5926832, 14 p. [CrossRef] [Google Scholar]
  • Song GJ, Wang QW, Chang HX (2011), Cramer rule for the unique solution of restricted matrix equations over the quaternion skew field. Comput Math Appl 61, 1576–1589. [Google Scholar]
  • Song G (2013), Characterization of the W-weighted Drazin inverse over the quaternion skew field with applications. Electr J Linear Algebra 26, 1–14. [Google Scholar]
  • Song GJ, Dong CZ (2017), New results on condensed Cramer’s rule for the general solution to some restricted quaternion matrix equations. J Appl Math Comput 53, 321–341. [Google Scholar]
  • Song GJ, Wang QW, Yu SW (2018), Cramer’s rule for a system of quaternion matrix equations with applications. Appl Math Comput 336, 490–499. [Google Scholar]
  • Aslaksen H (1996), Quaternionic determinants. Math Intell 18, 3, 57–65. [CrossRef] [Google Scholar]
  • Cohen N, De Leo S (2000), The quaternionic determinant. Electr J Linear Algebra 7, 100–111. [Google Scholar]
  • Chen L (1991), Definition of determinant and Cramer solutions over quaternion field. Acta Math Sin (N.S.) 7, 171–180. [CrossRef] [Google Scholar]
  • Dieudonné J (1943), Les determinantes sur un corps non-commutatif. Bull Soc Math France 71, 27–45. [CrossRef] [Google Scholar]
  • Dyson FJ (1972), Quaternion determinants. Helvetica Phys Acta 45, 289–302. [Google Scholar]
  • Gelfand I, Gelfand S, Retakh V, Wilson RL (2005), Quasideterminants. Adv Math 193, 56–141. [CrossRef] [Google Scholar]
  • Moore EH (1922), On the determinant of an Hermitian matrix of quaternionic elements. Bull Am Math Soc 28, 161–162. [Google Scholar]
  • Study E (1920), Zur Theorie der linearen Gleichungen. Acta Math 42, 1–61. [CrossRef] [Google Scholar]
  • Fan J (2003), Determinants and multiplicative functionals on quaternion matrices. Linear Algebra Appl 369, 193–201. [Google Scholar]
  • Wang QW, Li CK (2009), Ranks and the least-norm of the general solution to a system of quaternion matrix equations. Linear Algebra Appl 430, 1626–1640. [Google Scholar]