椭圆型交换四元数矩阵的实表示及逆矩阵求法
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摘要
在引入椭圆型交换四元数的基础上,首先证明了椭圆型交换四元数和实数域上的4阶矩阵是同构的,将对椭圆型交换四元数的研究转化为实数域上4阶矩阵的研究.其次,利用椭圆型交换四元数矩阵的实表示,将对椭圆型交换四元数矩阵的研究转化为实数域上4n阶矩阵的研究,得到了椭圆型交换四元数矩阵实表示的系列重要性质.最后,利用实表示的性质,得到椭圆型交换四元数矩阵特征值存在的充要条件,并给出椭圆型交换四元数矩阵逆矩阵的求法,且利用数值算例验证了结论的有效性.
Based on the introduction of elliptic commutative quaternion,firstly,it is proved that the elliptic commutative quaternion and the 4-order matrix on the real field are isomorphic.The study of the elliptic commutative quaternion is transformed into the study of the 4-order matrix on the real field.Secondly,using the real representation of the elliptic commutative quaternion matrix,the study of the elliptic commutative quaternion matrix is transformed into the study of the 4 n-order matrix on the real field.A series of important properties of real representation of elliptic commutative quaternion matrix are obtained.Finally,based on the real representation properties,the sufficient and necessary conditions for the existence of the eigenvalues of the elliptic commutative quaternion matrix are obtained.The method to find the inverse matrix of the elliptic commutative quaternion matrix is given.And the correctness of the result is verified by a numerical example.
Based on the introduction of elliptic commutative quaternion,firstly,it is proved that the elliptic commutative quaternion and the 4-order matrix on the real field are isomorphic.The study of the elliptic commutative quaternion is transformed into the study of the 4-order matrix on the real field.Secondly,using the real representation of the elliptic commutative quaternion matrix,the study of the elliptic commutative quaternion matrix is transformed into the study of the 4 n-order matrix on the real field.A series of important properties of real representation of elliptic commutative quaternion matrix are obtained.Finally,based on the real representation properties,the sufficient and necessary conditions for the existence of the eigenvalues of the elliptic commutative quaternion matrix are obtained.The method to find the inverse matrix of the elliptic commutative quaternion matrix is given.And the correctness of the result is verified by a numerical example.
引文
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[4]Alagoz Y,Oral K H,Yuce S.Split quaternion matrices[J].Miskolc Mathematical Notes,2012,13(2):223-232.
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[6]Ozdemir M.On eigenvalues of split quaternion matrix[J].Advances in Applied Clifford Algebras,2013,23(3):615-623.
[7]Jiang T S,Jiang Z W,Zhang Z Z.Algebraic techniques for diagonalization of a split quaternion matrix in split quaternionic mechanics[J].Journal of Mathematical Physics,2015,56(8):223-232.
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[11]Pinotsis D A.Segre quaternions,spectral analysis and a four-dimensional Laplace equation,in progress in analysis and its applications[M].Singapore:World Scientific,2010.
[12]Catoni F,Cannata R,Zampetti P.An introduction to constant curvature spaces in the commutative(segre)quaternion geometry[J].Advances in Applied Clifford Algebras,2006,16(1):85-101.
[13]Pei S C,Chang J H,Ding J J.Commutative reduced biquaternions and their fourier transform for signal and image processing applications[J].IEEE Transactions on Sinal Processing,2004,52(7):2012-2031.
[14]Catoni F,Cannata R,Zampetti P.An introduction to commutative quaternions[J].Advances in Applied Clifford Algebras,2006,16(1):1-28.
[15]Catoni F,Cannata R,Catoni V,et al.N-dimensional geometries generated by hypercomplex numbers[J].Advances in Applied Clifford Algebras,2005,15(1):1-25.
[16]Kosal H H,Tosun M.Commutative quaternion matrices[J].Advances in Applied Clifford Algebras,2014,16(3):769-799.
[2]Ozdemir M.The roots of split quaternion[J].Applied Mathematics Letters,2009,22(2):258-263.
[3]Kula L,Yayl Y.Split quaternions and rotations in semi Euclidean space[J].Journal of the Korean Mathematical Society,2007,44(6):1313-1327.
[4]Alagoz Y,Oral K H,Yuce S.Split quaternion matrices[J].Miskolc Mathematical Notes,2012,13(2):223-232.
[5]Zhang Z Z,Jiang Z W,Jiang T S.Algebraic methods for least squares problem in split quaternionic mechanics[J].Applied Mathematics and Computation,2015,269:618-625.
[6]Ozdemir M.On eigenvalues of split quaternion matrix[J].Advances in Applied Clifford Algebras,2013,23(3):615-623.
[7]Jiang T S,Jiang Z W,Zhang Z Z.Algebraic techniques for diagonalization of a split quaternion matrix in split quaternionic mechanics[J].Journal of Mathematical Physics,2015,56(8):223-232.
[8]Ozdemir M.On complex split quaternion matrices[J].Advances in Applied Clifford Algebras,2013,23(3):625-638.
[9]Segre C.The real representations of complex elements and extension to bicomplex systems[J].Mathematische Annalen,1892,40:413-467.
[10]Catoni F,Cannata R,Nichelatti E,et al.Commutative hypercomplex numbers and functions of hypercomplex variable:a matrix study[J].Advances in Applied Clifford Algebras,2005,15(2):183-213.
[11]Pinotsis D A.Segre quaternions,spectral analysis and a four-dimensional Laplace equation,in progress in analysis and its applications[M].Singapore:World Scientific,2010.
[12]Catoni F,Cannata R,Zampetti P.An introduction to constant curvature spaces in the commutative(segre)quaternion geometry[J].Advances in Applied Clifford Algebras,2006,16(1):85-101.
[13]Pei S C,Chang J H,Ding J J.Commutative reduced biquaternions and their fourier transform for signal and image processing applications[J].IEEE Transactions on Sinal Processing,2004,52(7):2012-2031.
[14]Catoni F,Cannata R,Zampetti P.An introduction to commutative quaternions[J].Advances in Applied Clifford Algebras,2006,16(1):1-28.
[15]Catoni F,Cannata R,Catoni V,et al.N-dimensional geometries generated by hypercomplex numbers[J].Advances in Applied Clifford Algebras,2005,15(1):1-25.
[16]Kosal H H,Tosun M.Commutative quaternion matrices[J].Advances in Applied Clifford Algebras,2014,16(3):769-799.
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