Acta mathematica scientia, Series B >
ESTIMATES OF THE DOUBLE DETERMINANTS OF QUATERNION MATRICES
Received date: 2007-06-08
Online published: 2010-07-20
Supported by
This research was supported in part by NCET (NCET06-9-23) and NUDT (JC08-02-03)
An estimate of the upper bound is given for the double determinant of the sum of two arbitrary quaternion matrices, and meanwhile the lower bound on the double determinant is established especially for the sum of two quaternion matrices which form an assortive pair. As applications, some known results are obtained as corollaries and a question in the matrix determinant theory is answered completely.
Key words： Assortive pair; double determinant; inequality; quaternion matrix
FENG Liang-Gui , CHENG Wei . ESTIMATES OF THE DOUBLE DETERMINANTS OF QUATERNION MATRICES[J]. Acta mathematica scientia, Series B, 2010 , 30(4) : 1189 -1198 . DOI: 10.1016/S0252-9602(10)60116-6
[1] Zhan X Z. Matrix Inequlities//Lecture Notes in Mathematics. Vol.1790. Berlin: Springer-Verlag Berlin Heidelberg, 2002
[2] Bruadli R A, Schneider H. Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley. Linear Algebra Appl, 1983, 52/53: 769--791
[3] Marcus M, Minc H. A Survey of Matrix Theory and Matrix Inequalities. Boston: Allyn and Bacon Inc, 1964
[4] Zhang F Z. Quaternions and matrices of quaternions. Linear Algebra Appl, 1997, 251: 21--57
[5] Zeng R Y. The quaternion matrix-valued Young's inequality. J Ineq Pure and Appl Math, 2005, 6(3): Art.89
[6] Zhuang W J. Inequalities of eigenvalues and singular values for quaternion matrices. Adv Math (China), 1988, 17(4): 403--407 (in Chinese)
[7] Cao C G. Some inequalities on trace of self-conjugate quaternion matrices. J of Math (PRC), 1988, 8(1): 313--314 (in Chinese)
[8] Lu Y X, Yuan Z Z, Zhang S Q. An inequality for singular values of product of quaternion matrices and its applications. J Math Res Expro, 1998, 18(3): 455--458 (in Chinese)
[9] Li W L. Some inequalities on the determinant of self-conjugate matrices over quaternion. J Inner Mongolia Normal Univ, Nat Sci, 1994, 1: 7--12 (in Chinese)
[10] Liu J Z. Matrices over quaternion division ring and theory of majorization. Acta Math Sin, 1992, 35(6): 831--838 (in Chinese)
[11] Deng H. Some matrix inequalities and estimates of the sectional curvatures of symmetric Riemannian spaces. Acta Math Sin, 1991, 34(3): 299--308 (in Chinese)
[12] Feng L G. The column left ranks of matrices over quaternion skew-field. Progress in Natural Science, 2006, 33(4): 222--227 (in Chinese)
[13] Li W L. Quaternion Matrices. Changsha: National University of Defense Technology Press, 2002 (in Chinese)
[14] Chen L X. Definition of determinant and Crammer solutions over the quaternion field. Acta Math Sin, New Series, 1991, 7(2): 171--180
[15] Chen L X. Inverse matrix and properties of double determinant over quaternion field. Sci China, 1991, 34A(5): 528--540
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