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A new simultaneous decomposition of a matrix quaternity over an arbitrary division ring with applications. (English) Zbl 1252.15014

The authors initially establish a simultaneous decomposition of a matrix quaternity over an arbitrary ring in which two of the matrices have the same number of columns while three of them have the same number of rows. In addition, they provide a practical algorithm for this decomposition. After, they present two systems of different type in order to apply their results. By using the explicit and easily testable conditions they give necessary and sufficient conditions for the existence of these systems. They also derive the general solution of these systems. After, they give two numerical examples to illustrate the results of the article. As a final conclusion it is mentioned that the findings of this article extend some known results in the literature.

MSC:

15A23 Factorization of matrices
15A03 Vector spaces, linear dependence, rank, lineability
15A21 Canonical forms, reductions, classification
15A22 Matrix pencils
15A24 Matrix equations and identities
12E15 Skew fields, division rings
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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References:

[1] Cai J., Math. Comput. Model. 50 pp 1237– (2009) · Zbl 1190.65061
[2] Chu D. L., Linear Algebra Appl. 311 pp 61– (2000) · Zbl 0970.65037
[3] Cohn P. M., Math. Z. 132 pp 151– (1973) · Zbl 0248.15010
[4] Cohn P. M., Skew-field Constructions, London Math. Soc. Lecture Notes 27 (1977) · Zbl 0355.16009
[5] Dehghan M., Appl. Math. Comput. 202 (2) pp 571– (2008) · Zbl 1154.65023
[6] Dehghan M., Comput. Math. Appl. 56 pp 3246– (2008) · Zbl 1165.15301
[7] Dehghan M., A 1 X 1 B 1 + A 2 X 2 B 2 = C. Math. Comput. Modelling 49 pp 1937– (2009)
[8] Guralnick R. M., Linear Algebra Appl. 39 pp 155– (1981) · Zbl 0468.16022
[9] Guralnick R. M., Linear Algebra Appl. 43 pp 125– (1982) · Zbl 0493.16015
[10] Gustafson W. H., Linear Algebra Appl. 231 pp 159– (1995) · Zbl 0840.15011
[11] Gustafson W., Linear Algebra Appl. 27 pp 219– (1979) · Zbl 0419.15009
[12] Hartwig R. E., Linear Algebra Appl. 39 pp 125– (1981) · Zbl 0467.15007
[13] Hungerford T. W., Algebra (1980) · Zbl 0442.00002
[14] Lam T. Y., J. Algebra 119 pp 308– (1988) · Zbl 0657.15015
[15] Li Y. T., Comput. Math. Appl. 55 pp 1142– (2008) · Zbl 1143.15012
[16] Lian D. Z., Algebra Colloq. 18 (2) pp 333– (2011)
[17] Mitra S. K., Proc. Cambridge Philos. Soc. 74 pp 213– (1973)
[18] Navarra A., Comput. Math. Appl. 41 pp 929– (2001) · Zbl 0983.15016
[19] Paige C. C., SIAM J. Number. Anal. 18 pp 398– (1981) · Zbl 0471.65018
[20] Pei S. C., IEEE International Symposium on Circuits and Systems pp 684– (1996)
[21] Semrl P., J. Algebra 298 (1) pp 142– (2006) · Zbl 1155.15302
[22] Sheng X. P., Appl. Math. Comput. 189 pp 1350– (2007) · Zbl 1133.65026
[23] Tian Y., Linear Multilinear Algebra 48 pp 123– (2000) · Zbl 0970.15005
[24] Tian Y., J. Statist. Plann. Inference 139 pp 2445– (2009) · Zbl 1160.62051
[25] vander Woude J. W., Systems Control Lett. 9 pp 7– (1987) · Zbl 0623.93028
[26] vander Woude , J. W. ( 1987 ). Feedback decoupling and stabilization for linear system with multiple exogenous variables. Ph.D. Thesis, Technical Univ. of Eindhoven, Notherlands .
[27] Van Loan C. F., SIAM J. Number. Anal. 13 pp 76– (1976) · Zbl 0338.65022
[28] Wang Q. W., Linear Algebra Appl. 384 pp 43– (2004) · Zbl 1058.15015
[29] Wang Q. W., Acta Math. Sinica, English Series 21 (2) pp 323– (2005) · Zbl 1083.15021
[30] Wang Q. W., Comput. Math. Appl. 49 pp 665– (2005) · Zbl 1138.15004
[31] Wang Q. W., Comput. Math. Appl. 49 pp 641– (2005) · Zbl 1138.15003
[32] Wang Q. W., Algebra Colloq. 14 (4) pp 555– (2007) · Zbl 1143.15013
[33] Wang Q. W., Electron. J. Linear Algebra 20 pp 552– (2010)
[34] Wang Q. W., Linear Algebra Appl. 430 pp 1626– (2009) · Zbl 1158.15010
[35] Wang Q. W., Algebra Colloq. 16 (2) pp 293– (2009) · Zbl 1176.15020
[36] Wang Q. W., Linear Algebra Appl. 353 pp 169– (2002) · Zbl 1004.15017
[37] Wang Q. W., Linear Algebra Appl. 431 pp 2291– (2009) · Zbl 1180.15019
[38] DOI: 10.1007/s11425-009-0044-4 · Zbl 1193.53167
[39] Wang Q. W., Comm. Algebra 37 (6) pp 2060– (2009) · Zbl 1393.15021
[40] Wang Q. W., Algebra Colloq. 17 (2) pp 345– (2010) · Zbl 1188.15016
[41] Wang Q. W., Electron. J. Linear Algebra 17 pp 88– (2008)
[42] Wang Q. W., Electron. J. Linear Algebra. 18 pp 289– (2009)
[43] Wiegmann N. A., Canada J. Math. 7 pp 191– (1955) · Zbl 0064.01604
[44] Xie D., J. Comput. Appl. Math 216 pp 484– (2008) · Zbl 1138.74050
[45] Yuan S., Math. Comput. Model. 48 pp 91– (2008) · Zbl 1145.15303
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