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Tytuł artykułu

Consimilarity and quaternion matrix equations AX −^X B = C, X − A^X B = C

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Języki publikacji
EN
Abstrakty
EN
L. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30] gave a canonical form of a quaternion matrix with respect to consimilarity transformations A ↦ ˜S−1AS in which S is a nonsingular quaternion matrix and h = a + bi + cj + dk ↦ ˜h := a − bi + cj − dk (a, b, c, d ∈ ℝ). We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformations A ↦^S−1AS in which h ↦ ^h is an arbitrary involutive automorphism of the skew field of quaternions. We apply the obtained canonical form to the quaternion matrix equations AX −^X B = C and X − A^X B = C.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
otrzymano
2014-06-12
zaakceptowano
2014-10-08
online
2014-12-01
Twórcy
  • Kiev National Taras Shevchenko University, Kiev, Ukraine
  • Institute of Mathematics, Tereshchenkivska 3, Kiev, Ukraine
Bibliografia
  • [1] J.H. Bevis, F.J. Hall, R.E. Hartwig, Consimilarity and thematrix equation A¯X −XB = C, Current Trends inMatrix Theory (Auburn,Ala., 1986), North-Holland, New York, 1987, pp. 51–64.
  • [2] H. Dym, Linear Algebra in Action, American Mathematical Society, 2007.
  • [3] F.R. Gantmacher, The Theory of Matrices, Vol. 1, Chelsea, New York, 1959.
  • [4] Y.P. Hong, R.A. Horn, A canonical form for matrices under consimilarity, Linear Algebra Appl. 102 (1988) 143–168.
  • [5] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013.
  • [6] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
  • [7] L. Huang, Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331 (2001) 21–30.
  • [8] N. Jacobson, The Theory of Rings, American Mathematical Society, New York, 1943.
  • [9] T. Jiang, M. Wei, On solutions of the matrix equations X − AXB = C and X − A¯X B = C, Linear Algebra Appl. 367 (2003)225–233.
  • [10] T.S. Jiang, M.S. Wei, On a solution of the quaternion matrix equation X − A˜X B = C and its application, Acta Math. Sin. 21(2005) 483–490.
  • [11] A.W. Knapp, Advanced Algebra, Birkhaüser, 2007.
  • [12] P. Lancaster, Explicit solutions of linear matrix equations, SIAM Review 12 (1970) 544–566.[WoS][Crossref]
  • [13] P. Lancaster, L. Lerer, Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices, Linear AlgebraAppl. 62 (1984) 19–49.
  • [14] P. Lancaster, M. Tismenetsky, The Theory of Matrices with Applications, 2nd ed., Academic Press, 1985.
  • [15] C. Song, G. Chen, On solutions of matrix equation XF − AX = C and XF − A˜X = C over quaternion field, J. Appl. Math.Comput. 37 (2011) 57–68.
  • [16] C.Q. Song, G.L. Chen, Q.B. Liu, Explicit solutions to the quaternion matrix equations X − AXF = C and X − A˜XF = C, Int. J.Comput. Math. 89 (2012) 890–900.
  • [17] C. Song, J. Feng, X.Wang, J. Zhao, A real representation method for solving Yakubovich-j-conjugate quaternionmatrix equation,Abstr. Appl. Anal. 2014, Art. ID 285086, 9 pp.
  • [18] M.-F. Vignéras, Arithmétique des Algèbres de Quaternions, Springer, Berlin, 1980.
  • [19] N.A. Wiegmann, Some theorems on matrices with real quaternion elements, Canad. J. Math. 7 (1955) 191–201.
  • [20] S.F. Yuan, A.P. Liao, Least squares solution of the quaternion matrix equation X − A^XB = C with the least norm, LinearMultilinear Algebra 59 (2011) 985–998.[WoS]
  • [21] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997) 21–57.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_spma-2014-0018
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