Abstract
We present some practical necessary and sufficient conditions for the existence of an \(\eta \)-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation. We also provide the general \(\eta \)-(anti-)Hermitian solution to the constrained matrix equation when it is solvable. Moreover, we present algorithms and numerical examples to illustrate the results of this paper.
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The data underlying this article will be shared on reasonable request to the corresponding author [Qing-Wen, Wang].
References
Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. International Series of Monographs on Physics, vol. 88. The Clarendon Press, Oxford University Press, New York (1995)
Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications. John Wiley and Sons, New York (1974)
Catoni, F.: Commutative (Segre’s) quaternion fields and relation with Maxwell equations. Adv. Appl. Clifford Algebras 18, 9–28 (2008)
Cyrus, J., Clive, C.T., Danilo, P.M.: A class of quaternion valued affine projection algorithms. Signal Process. 93(7), 1712–1723 (2013)
Guo, L., Zhu, M., Ge, X.: Reduced biquaternion canonical transform, convolution and correlation. Signal Process. 91, 2147–2153 (2011)
Hamilton, W.R.: Lectures on Quaternions. Hodges and Smith, Dublin (1853)
Isokawa, T., Nishimura, H., Matsui, N.: Commutative quaternion and multistate Hopfield neural networks. The 2010 International Joint Conference on Neural Networks (IJCNN). Barcelona, Spain, 1–6 (2010)
Khatri, C.G., Mitra, S.K.: Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J. Appl. Math. 31, 579–585 (1976)
Kösal, H.H., Akyig̈it, M.: Consimilarity of commutative quaternion matrices. Miskolc Math. Notes 16, 965–977 (2015)
Kösal, H.H., Tosun, M.: Commutative quaternion matrices. Adv. Appl. Clifford Algebras 24, 769–779 (2014)
Kösal, H.H., Tosun, M.: Universal similarity factorization equalities for commutative quaternions and their matrices. Linear Multilinear Algebra 67, 926–938 (2019)
Liu, X., He, Z.H.: On the split quaternion matrix equation \(AX=B\). Banach J. Math. Anal. 14, 228–248 (2020)
Liu, X., Zhang, Y.: Least-squares solutions \(X =X^{\eta * }\) to split quaternion matrix equation \(AXA^{\eta *} = B\). Math. Methods Appl. Sci. 43, 2189–2201 (2019)
Liu, X., Wang, Q.W., Zhang, Y.: Consistency of quaternion matrix equations \(AX^{*} -XB = C\) and \(X- AX^{*}B = C\). Electron. Linear Algebra 35, 394–407 (2019)
Miao, J.F., Kou, K.I.: Color image recovery using low-rank quaternion matrix completion algorithm. IEEE Trans. Image Process. 31, 190–201 (2022)
Özen, K.E., Tosun, M.: On the matrix algebra of elliptic biquaternions. Math. Methods Appl. Sci. 43(6), 2984–2998 (2020)
Pei, S.C., Chang, J.H., Ding, J.J.: Commutative reduced biquaternions and their Fourier transform for signal and image processing applications. IEEE Trans. Signal Process. 52, 2012–2031 (2004)
Ren, B.Y., Wang, Q.W., Chen, X.Y.: The \({\eta }\)-anti-Hermitian solution to a constrained matrix equation over the generalized Segre quaternion algebra. Symmetry 15(3), 592 (2023)
Segre, C.: The real representations of complex elements and extension to bicomplex systems. Math. Ann. 40, 413–467 (1892)
Tian, Y., Liu, X., Zhang, Y.: Least-squares solutions of the generalized reduced biquaternion matrix equations. Filomat 37, 863–870 (2023)
Took, C.C., Mandic, D.P.: Quaternion-valued stochastic gradient-based adaptive IIR filtering. IEEE Trans. Signal Process. 58, 3895–3901 (2010)
Took, C.C., Mandic, D.P.: Augmented second-order statistics of quaternion random signals. Signal Process. 91, 214–224 (2011)
Yuan, S.F., Wang, Q.W., Yu, Y.B., Tian, Y.: On Hermitian solutions of the split quaternion matrix equation \(AXB + CXD = E\). Adv. Appl. Clifford Algebras 27(4), 3235–3252 (2017)
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This research was supported by the grant from the National Natural Science Foundation of China (11971294).
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Communicated by Fuzhen Zhang.
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Chen, XY., Wang, QW. The \(\eta \)-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation. Banach J. Math. Anal. 17, 40 (2023). https://doi.org/10.1007/s43037-023-00262-5
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DOI: https://doi.org/10.1007/s43037-023-00262-5