Existence of flipped orthogonal conjugate symmetric Jordan canonical bases for real H-selfadjoint matrices

Abstract

For real matrices selfadjoint in an indefinite inner product there are two special canonical Jordan forms, that is (i) flipped orthogonal (FO) and (ii) γ-conjugate symmetric (CS). These are the classical Jordan forms with certain additional properties induced by the fact that they are H-selfadjoint. In this paper, we prove that for any real H-selfadjoint matrix, there is a γ-FOCS Jordan form that is simultaneously flipped orthogonal and γ-conjugate symmetric.

Keywords:

• Canonical Jordan bases
• indefinite inner product
• H-selfadjoint matrices

2010 Mathematics Subject Classifications:

• 15A20
• 15A21
• 46C20

Acknowledgments

The authors are indebted to the anonymous referee for the careful reading of this manuscript, pointing out numerous typos, and giving suggestions that helped to improve the presentation of the material.

Disclosure statement

No potential conflict of interest was reported by the author(s).