Trace Invariance for Quaternion Matrices

  • Ralph John de la Cruz University of the Philippines Diliman

Abstract

Let F be a f ield. It is a classical result in linear algebra that for each A, P ϵ Mn (F) such that P is nonsingular, tr A = tr (PAP-1). We show in this paper that the preceding property does not hold true if F is the division ring of real quaternions. We show that the only quaternion matrices that have their trace invariant under unitary similarity are Hermitian matrices, and that the only matrices that have their trace invariant under similarity are real scalar matrices.

Keywords: Trace, quaternion, unitary matrices, Hermitian matrices

LAYMAN’S ABSTRACT

We consider a classical result in linear algebra concerning the trace of
matrices with complex entries and we determine whether this result
holds true for the set of quaternion matrices.

Published
2015-12-15
How to Cite
DE LA CRUZ, Ralph John. Trace Invariance for Quaternion Matrices. Science Diliman: A Journal of Pure and Applied Sciences, [S.l.], v. 27, n. 2, dec. 2015. ISSN 2012-0818. Available at: <https://journals.upd.edu.ph/index.php/sciencediliman/article/view/4806>. Date accessed: 20 july 2025.
Citation Formats
Issue
Vol 27 No 2 (2015)
Section
Articles
Submission of a manuscript implies: that the work described has not been published before (except in the form of an abstract or as part of a published lecture, review, or thesis); that it is not under consideration for publication elsewhere; that its publication has been approved by all co-authors, if any, as well as by the responsible authorities at the institute where the work has been carried out; that, if and when the manuscript is accepted for publication, the authors agree to the automatic transfer of the copyright to the publisher; that the manuscript will not be published elsewhere in any language without the consent of the copyright holders; that written permission of the copyright holder is obtained by the authors for material used from other copyrighted sources; and that any costs associated with obtaining this permission are the authors’ responsibility.