Abstract
The complex matrix equation \(A\overline{X}-XB=C\) is of interest in simplifying matrix representations of semilinear transformations arising from quantum mechanics. However, there has not been a complete algorithm for solving it so far. In this paper, we provide a complete algorithm to compute solutions of the complex matrix equation \(A\overline{X}-XB=C\), and give an example. The results of this paper manifest the advantage of the quaternion matrix theory in solving some complex matrix problems.
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References
Bevis, J.H., Hall, F.J., Hartwig, R.E.: The matrix equation \(A\overline{X}-XB=C\) and its special cases. SIAM J. Matrix Anal. Appl. 19, 348–359 (1988)
Truhar, N., Tomljanović, Z., Li, R.C.: Analysis of the solution of the Sylvester equation using low-rank ADI with exact shifts. Syst. control Lett. 59, 248–257 (2010)
Chen, C., Schonfeld, D.: Pose estimation from multiple cameras based on Sylvester’s equation. Comput. Vis. Image Und. 114, 652–666 (2010)
Halabi, S., Souley Ali, H., Rafaralahy, H., Zasadzinski, M.: \(\mathcal{H}_{\infty}\) functional filtering for stochastic bilinear systems with multiplicative noises. Automatica 45, 1038–1045 (2010)
Bevis, J.H., Hall, F.J., Hartwig, R.E.: Consimilarity and the matrix equation \(A\overline{X}-XB=C\). In: Uhlig, F., Grone, R. (eds.) Current Trends in Matrix Theory, pp. 51–64. North-Holland, Amsterdam (1987)
Wu, A.G., Duan, G.R., Yu, H.H.: On solutions of the matrix equations XF − AX = C and \(XF-A\overline{X}=C\). Appl. Math. Comput. 183, 932–941 (2006)
Wu, A.G., Fu, Y.M., Duan, G.R.: On solutions of matrix equations V − AVF = BW and \(V-A\overline{V}F=BW\). Math. Comput. Model. 47, 1181–1197 (2008)
Wu, A.G., Wang, H.Q., Duan, G.R.: On matrix equations X − AXF = C and \(X-A\overline{X}F=C\). Appl. Math. Comput. 230, 690–698 (2009)
Huang, L.P.: Consimilarity of quaternion matrices and complex matrices. Linear Algebra Appl. 331, 21–30 (2001)
Wiegmann, N.A.: Some theorems on matrices with real quaternion elements. Cana. J. Math. 7, 191–201 (1955)
Zhang, F.Z., Wei, Y.: Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices. Comm. Algebra 29, 2363–2375 (2001)
Huang, L.P.: Jordan canonical form of a matrix over quaternion field. Northeast. Math. J. 10, 18–24 (1994)
Huang, L.P.: The matrix equation AXB − GXD = E over quaternion field. Linear Algebra Appl. 234, 197–208 (1996)
Feng, L.G., Cheng, W.: Algorithm for solving the transition matrix of Jordan canonical form over quaternion skew-field (submitted for publication)
Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)
Quaternion Toolbox for Matlab ®, [online], software library, written by Stephen Sangwine and Nicolas Le Bihan, http://qtfm.sourceforge.net/
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Yu, S., Cheng, W., feng, L. (2010). Algorithm for Solving the Complex Matrix Equation \(A\overline{X}-XB=C\) . In: Zhu, R., Zhang, Y., Liu, B., Liu, C. (eds) Information Computing and Applications. ICICA 2010. Communications in Computer and Information Science, vol 105. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16336-4_10
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DOI: https://doi.org/10.1007/978-3-642-16336-4_10
Publisher Name: Springer, Berlin, Heidelberg
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