×

Polar decompositions of quaternion matrices in indefinite inner product spaces. (English) Zbl 1482.15026

Summary: Polar decompositions of quaternion matrices with respect to a given indefinite inner product are studied. Necessary and sufficient conditions for the existence of an \(H\)-polar decomposition are found. In the process, an equivalent to Witt’s theorem on extending \(H\)-isometries to \(H\)-unitary matrices is given for quaternion matrices.

MSC:

15B33 Matrices over special rings (quaternions, finite fields, etc.)
47B50 Linear operators on spaces with an indefinite metric
15A23 Factorization of matrices
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] D. Alpay, A.C.M. Ran, and L. Rodman. Basic classes of matrices with respect to quaternionic indefinite inner product spaces.Linear Algebra Appl.,416:242-269, 2006. · Zbl 1102.15014
[2] Y. Bolshakov, C.V.M. van der Mee, A.C.M. Ran, B. Reichstein, and L. Rodman. Polar decompositions in finite dimensional indefinite scalar product spaces: General theory.Linear Algebra Appl.,216:91-141, 1997. · Zbl 0881.15013
[3] Y. Bolshakov, C.V.M. van der Mee, A.C.M. Ran, B. Reichstein, and L. Rodman. Polar decompositions in finite dimensional indefinite scalar product spaces: Special cases and applications.Recent Develop. Oper. Theory Appl., 87:61-94, 1996. · Zbl 0861.15012
[4] Y. Bolshakov, C.V.M. van der Mee, A.C.M. Ran, B. Reichstein, and L. Rodman. Errata for: Polar decompositions in finite dimensional indefinite scalar product spaces: Special cases and applications.Integr. Equ. Oper. Theory, 27:497-501, 1997. · Zbl 0897.15005
[5] Y. Bolshakov, C.V.M. van der Mee, A.C.M. Ran, B. Reichstein, and L. Rodman.Extension of isometries in finitedimensional indefinite scalar product spaces and polar decompositions.Linear Algebra Appl.,18(3):752-774, 1997. · Zbl 0874.15022
[6] Y. Bolshakov and B. Reichstein.Unitary equivalence in an indefinite scalar product: An analogue of singular-value decomposition.Linear Algebra Appl.,222:155-226, 1995. · Zbl 0828.15009
[7] G.J. Groenewald, D.B. Janse van Rensburg, A.C.M. Ran, F. Theron, and M. van Straaten.mth roots ofH-selfadjoint matrices.Linear Algebra Appl.,610:804-826, 2021. · Zbl 1460.15015
[8] D.B. Janse van Rensburg, A.C.M. Ran, F. Theron, and M. van Straaten.mth roots ofH-selfadjoint matrices over the quaternions.Electron. J. Linear Algebra,Preprint, To appear: 2021. · Zbl 1471.15025
[9] M. Karow. Self-adjoint operators and pairs of Hermitian forms over the quaternions.Linear Algebra Appl.,299:101-117, 1999. · Zbl 0951.15015
[10] C.V.M. van der Mee, A.C.M. Ran, and L. Rodman. Stability of self-adjoint square roots and polar decompositions in indefinite scalar product spacesLinear Algebra Appl., 302-303:77-104, 1999. · Zbl 0954.15008
[11] L. Rodman.Topics in Quaternion Linear Algebra. Princeton University Press, Princeton, 2014. · Zbl 1304.15004
[12] N.A. Wiegmann. Some theorems on matrices with real quaternion elements.Canad. J. Math.,7:191-201, 1955. · Zbl 0064.01604
[13] F. Zhang. Quaternions and Matrices of Quaternions.Linear Algebra Appl.,251:21-57, 1997. · Zbl 0873.15008
[14] F. Zhang and Y. Wei. Jordan canonical form of a partitioned complex matrix and its application to real quaternion matrices.Comm. Algebra,29:2363-2375, 2001 · Zbl 0999.15015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.