Unifying unitary and hyperbolic transformations
type: Item
instance of: scholarly article
title: Unifying unitary and hyperbolic transformations
zbMATH Open document ID: 0970.65042
DOI: 10.1016/S0024-3795(00)00108-7
author: Sanzheng Qiao
author: Allan O. Steinhardt
author: Adam W. Bojanczyk
published in: Linear Algebra and its Applications
publication date: 2001-10-07T00:00:00Z
review text: Generalized unitary matrices are studied and described in known unified formulas with respect to arbitrary (in-)definite inner products. Here \(V\) is called \(S=\) diag\((\pm 1)\) unitary if \(V^*SV=S\) for the signature matrix \(S\). The unified transformations are applied to obtain generalized Cholesky factorizations of strongly nonsingular indefinite Hermitian matrices. Their condition numbers are studied with an eye on proper pivoting strategies. Multiplying by generalized Householder matrices is shown to be relationally stable. However, this paper does not mention any of the previous occurences and uses of generalized orthogonal matrices such as in algebraic group theory or in numerical matrix eigenvalue algorithms, as in the (unstable) HR algorithm [cf. \textit{A. Burse-Tertner}, ibid. 35, 155-173 (1981; Zbl 0462.65021)] or the DQR algorithm [cf. \textit{F. Uhlig}, Numer. Math. 76, No.~4, 515-553 (1997; Zbl 0883.65029)].
Mathematics Subject Classification ID: 65F25
Mathematics Subject Classification ID: 15A12
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1533000
zbMATH Keywords: Householder transform
zbMATH Keywords: generalized unitary matrix
zbMATH Keywords: hyperbolic transformation
zbMATH Keywords: reflection
zbMATH Keywords: generalized rotation
zbMATH Keywords: generalized Cholesky factorization
zbMATH Keywords: signature matrix
zbMATH Keywords: condition numbers
zbMATH Keywords: pivoting strategies
zbMATH Keywords: eigenvalue algorithms
reviewed by: Frank Uhlig
describes a project that uses: LINPACK
MaRDI profile type: Publication
cites work: A Note on Downdating the Cholesky Factorization
cites work: Stability Analysis of a Householder-Based Algorithm for Downdating the Cholesky Factorization
cites work: Hyperbolic Householder Algorithms for Factoring Structured Matrices
cites work: Matrices and indefinite scalar products
cites work: Q5689624
cites work: Hyperbolic householder transformations
cites work: Hyperbolic Householder Transforms
cites work: The Effects of Rounding Error on an Algorithm for Downdating a Cholesky Factorization
cites work: On Hyperbolic Triangularization: Stability and Pivoting
Wikidata QID: Q126591688
full work available at URL: s0024-3795(00)00108-7
OpenAlex ID: W1995386581
description: scientific article; zbMATH DE number 1533000