Canonical form for H-symplectic matrices

G. J. Groenewald; D. B. Janse van Rensburg; A. C.M. Ran

doi:10.1007/978-3-030-04269-1_11

Canonical form for H-symplectic matrices

G. J. Groenewald^*, D. B. Janse van Rensburg, A. C.M. Ran

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

175 Downloads (Pure)

Abstract

In this paper we consider pairs of matrices (A,H), with A and H either both real or both complex, H is invertible and skew-symmetric and A is H -symplectic, that is, A^TH A = H. A canonical form for such pairs is derived under the transformations (A,H) → (S ⁻¹AS, S^TH S) for invertible matrices S. In the canonical form for the pair, the matrix A is brought in standard (real or complex) Jordan normal form, in contrast to existing canonical forms.

Original language	English
Title of host publication	Operator Theory, Analysis and the State Space Approach
Subtitle of host publication	In Honor of Rien Kaashoek
Editors	Harm Bart, Sanne ter Horst, André C.M. Ran, Hugo J. Woerdeman
Publisher	Springer International Publishing Switzerland
Pages	269-290
Number of pages	22
ISBN (Electronic)	9783030042691
ISBN (Print)	9783030042684
DOIs	https://doi.org/10.1007/978-3-030-04269-1_11
Publication status	Published - 2018

Publication series

Name	Operator Theory: Advances and Applications
Volume	271
ISSN (Print)	0255-0156
ISSN (Electronic)	2296-4878

Keywords

Canonical forms
H -symplectic matrices
Indefinite inner product space

Access to Document

10.1007/978-3-030-04269-1_11

Groenewald2018_Chapter_CanonicalFormForH-symplecticMatricesFinal published version, 434 KBLicence: Article 25fa Dutch Copyright Act

Persistent URL (handle)

Persistent link

Cite this

Groenewald, G. J., Janse van Rensburg, D. B., & Ran, A. C. M. (2018). Canonical form for H-symplectic matrices. In H. Bart, S. ter Horst, A. C. M. Ran, & H. J. Woerdeman (Eds.), Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek (pp. 269-290). (Operator Theory: Advances and Applications; Vol. 271). Springer International Publishing Switzerland. https://doi.org/10.1007/978-3-030-04269-1_11

Groenewald, G. J. ; Janse van Rensburg, D. B. ; Ran, A. C.M. / Canonical form for H-symplectic matrices. Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. editor / Harm Bart ; Sanne ter Horst ; André C.M. Ran ; Hugo J. Woerdeman. Springer International Publishing Switzerland, 2018. pp. 269-290 (Operator Theory: Advances and Applications).

@inbook{e41c1dda120e4d60aeeece5f396d5642,

title = "Canonical form for H-symplectic matrices",

abstract = "In this paper we consider pairs of matrices (A,H), with A and H either both real or both complex, H is invertible and skew-symmetric and A is H -symplectic, that is, ATH A = H. A canonical form for such pairs is derived under the transformations (A,H) → (S −1AS, STH S) for invertible matrices S. In the canonical form for the pair, the matrix A is brought in standard (real or complex) Jordan normal form, in contrast to existing canonical forms.",

keywords = "Canonical forms, H -symplectic matrices, Indefinite inner product space",

author = "Groenewald, {G. J.} and {Janse van Rensburg}, {D. B.} and Ran, {A. C.M.}",

year = "2018",

doi = "10.1007/978-3-030-04269-1_11",

language = "English",

isbn = "9783030042684",

series = "Operator Theory: Advances and Applications",

publisher = "Springer International Publishing Switzerland",

pages = "269--290",

editor = "Harm Bart and {ter Horst}, Sanne and Ran, {Andr{\'e} C.M.} and Woerdeman, {Hugo J.}",

booktitle = "Operator Theory, Analysis and the State Space Approach",

}

Groenewald, GJ, Janse van Rensburg, DB & Ran, ACM 2018, Canonical form for H-symplectic matrices. in H Bart, S ter Horst, ACM Ran & HJ Woerdeman (eds), Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. Operator Theory: Advances and Applications, vol. 271, Springer International Publishing Switzerland, pp. 269-290. https://doi.org/10.1007/978-3-030-04269-1_11

Canonical form for H-symplectic matrices. / Groenewald, G. J.; Janse van Rensburg, D. B.; Ran, A. C.M.
Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. ed. / Harm Bart; Sanne ter Horst; André C.M. Ran; Hugo J. Woerdeman. Springer International Publishing Switzerland, 2018. p. 269-290 (Operator Theory: Advances and Applications; Vol. 271).

Research output: Chapter in Book / Report / Conference proceeding › Chapter › Academic › peer-review

TY - CHAP

T1 - Canonical form for H-symplectic matrices

AU - Groenewald, G. J.

AU - Janse van Rensburg, D. B.

AU - Ran, A. C.M.

PY - 2018

Y1 - 2018

N2 - In this paper we consider pairs of matrices (A,H), with A and H either both real or both complex, H is invertible and skew-symmetric and A is H -symplectic, that is, ATH A = H. A canonical form for such pairs is derived under the transformations (A,H) → (S −1AS, STH S) for invertible matrices S. In the canonical form for the pair, the matrix A is brought in standard (real or complex) Jordan normal form, in contrast to existing canonical forms.

AB - In this paper we consider pairs of matrices (A,H), with A and H either both real or both complex, H is invertible and skew-symmetric and A is H -symplectic, that is, ATH A = H. A canonical form for such pairs is derived under the transformations (A,H) → (S −1AS, STH S) for invertible matrices S. In the canonical form for the pair, the matrix A is brought in standard (real or complex) Jordan normal form, in contrast to existing canonical forms.

KW - Canonical forms

KW - H -symplectic matrices

KW - Indefinite inner product space

UR - http://www.scopus.com/inward/record.url?scp=85060166112&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060166112&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-04269-1_11

DO - 10.1007/978-3-030-04269-1_11

M3 - Chapter

AN - SCOPUS:85060166112

SN - 9783030042684

T3 - Operator Theory: Advances and Applications

SP - 269

EP - 290

BT - Operator Theory, Analysis and the State Space Approach

A2 - Bart, Harm

A2 - ter Horst, Sanne

A2 - Ran, André C.M.

A2 - Woerdeman, Hugo J.

PB - Springer International Publishing Switzerland

ER -

Groenewald GJ, Janse van Rensburg DB, Ran ACM. Canonical form for H-symplectic matrices. In Bart H, ter Horst S, Ran ACM, Woerdeman HJ, editors, Operator Theory, Analysis and the State Space Approach: In Honor of Rien Kaashoek. Springer International Publishing Switzerland. 2018. p. 269-290. (Operator Theory: Advances and Applications). Epub 2018 Dec 31. doi: 10.1007/978-3-030-04269-1_11

Canonical form for H-symplectic matrices

Abstract

Publication series

Keywords

Access to Document

Persistent URL (handle)

Other files and links

Fingerprint

Cite this