Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices. (English) Zbl 1482.15016
The study of quaternion determinants dates back to the work of A. Cayley [“On certain results relating to quaternions”, London Edinburgh Dublin Philos. Mag. J. Sci. 26, No. 171, 141–145 (1845)]. The abstract approach to the problem of the computation of the determinant of a matrix with non-commutative elements leads to various notions for determinants. In the present paper the authors follow the definition in [B. Xie, Acta Math. Sin. 23, 668–683 (1980; Zbl 0466.15007)] of the determinants for self-adjoint matrices over the quaternions. The authors provide determinantal inequalities for positive definite quaternion matrices extending the usual Hua-Marcus-Zhang-type inequalities for positive definite matrices with complex coefficients. The main goal is to obtain an inequality involving the eigenvalues and the singular values using a the generalized Schur decomposition for quaternion matrices. The extension of the results by L.-K. Hua [Acta Math. Sin. 5, 463–470 (1955; Zbl 0066.26601)], M. Marcus [Am. Math. Mon. 65, 266–269 (1958; Zbl 0083.00802)] and F. Zhang [Matrix theory. Basic results and techniques. 2nd ed. New York, NY: Springer (2011; Zbl 1229.15002), Theorem 7.18] follow by a direct application of the aforementioned inequality.
Reviewer: Hassan Issa (Beirut)
MSC:
| 15A45 | Miscellaneous inequalities involving matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
| 20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |
Keywords:
determinantal inequality; matrix of quaternions; eigenvalue; singular value; positive definite matrix; Hua-Marcus-Zhang typeReferences:
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