Some Properties of the Exchange Operator with respect to Structured Matrices defined by Indefinite Scalar Product Spaces

2015 ◽  
Vol 30 ◽  
Author(s):  
Hanz Martin Cheng ◽  
Roden Jason David
Keyword(s):  
Scalar Product ◽  
Structured Matrices ◽  
Product Spaces ◽  
Orthogonal Matrices ◽  
Structured Matrix ◽  
Exchange Operator ◽  
Siam Review

The properties of the exchange operator on some types of matrices are explored in this paper. In particular, the properties of exc(A,p,q), where A is a given structured matrix of size (p+q)×(p+q) and exc : M ×N×N → M is the exchange operator are studied. This paper is a generalization of one of the results in [N.J. Higham. J-orthogonal matrices: Properties and generation. SIAM Review, 45:504–519, 2003.].

scholarly journals Sign patterns of J-orthogonal matrices

2017 ◽  
Vol 5 (1) ◽  
pp. 225-241
Author(s):  
Frank J. Hall ◽  
Zhongshan Li ◽  
Caroline T. Parnass ◽  
Miroslav Rozložník
Keyword(s):  
Orthogonal Matrix ◽  
Orthogonal Matrices ◽  
Sign Patterns ◽  
Exchange Operator ◽  
Czechoslovak Math ◽  
Siam Review ◽  
G Matrices ◽  
Block Diagonal

Abstract This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns allow J-orthogonality. Important tools in this analysis are Theorem 2.2 on the exchange operator and Theorem 3.2 on the characterization of J-orthogonal matrices in the paper “J-orthogonal matrices: properties and generation", SIAM Review 45 (3) (2003), 504-519, by Higham. As a result, it follows that for n ≤4 all n×n full sign patterns allow a J-orthogonal matrix as well as a G-matrix. In addition, the 3 × 3 sign patterns of the J-orthogonal matrices which have zero entries are characterized.

scholarly journals G-reflectors: analogues of Householder transformations in scalar product spaces

2004 ◽  
Vol 385 ◽  
pp. 187-213 ◽  
Author(s):  
D.Steven Mackey ◽  
Niloufer Mackey ◽  
Françoise Tisseur
Keyword(s):  
Scalar Product ◽  
Product Spaces ◽  
Householder Transformations

scholarly journals Stability of self-adjoint square roots and polar decompositions in indefinite scalar product spaces

1999 ◽  
Vol 302-303 ◽  
pp. 77-104 ◽  
Author(s):  
Cornelis V.M. van der Mee ◽  
André C.M. Ran ◽  
Leiba Rodman
Keyword(s):  
Scalar Product ◽  
Product Spaces ◽  
Square Roots

scholarly journals A Note on the vec Operator Applied to Unbalanced Block-Structured Matrices

2016 ◽  
Vol 2016 ◽  
pp. 1-3 ◽  
Author(s):  
Hal Caswell ◽  
Silke F. van Daalen
Keyword(s):  
Simple Formula ◽  
Column Vector ◽  
Structured Matrices ◽  
Structured Matrix ◽  
Block Structured

The vec operator transforms a matrix to a column vector by stacking each column on top of the next. It is useful to write the vec of a block-structured matrix in terms of the vec operator applied to each of its component blocks. We derive a simple formula for doing so, which applies regardless of whether the blocks are of the same or of different sizes.

scholarly journals Polar decompositions in finite dimensional indefinite scalar product spaces: General theory

1997 ◽  
Vol 261 (1-3) ◽  
pp. 91-141 ◽  
Author(s):  
Yuri Bolshakov ◽  
Cornelis V.M. van der Mee ◽  
AndréC.M. Ran ◽  
Boris Reichstein ◽  
Leiba Rodman
Keyword(s):  
General Theory ◽  
Scalar Product ◽  
Product Spaces ◽  
Finite Dimensional

Polar Decompositions in Finite Dimensional Indefinite Scalar Product Spaces: Special Cases and Applications

1996 ◽  
pp. 61-94 ◽  
Author(s):  
Yuri Bolshakov ◽  
Cornelis V. M. van der Mee ◽  
André C. M. Ran ◽  
Boris Reichstein ◽  
Leiba Rodman
Keyword(s):  
Scalar Product ◽  
Product Spaces ◽  
Finite Dimensional ◽  
Special Cases

scholarly journals Structured Factorizations in Scalar Product Spaces

2005 ◽  
Vol 27 (3) ◽  
pp. 821-850 ◽  
Author(s):  
D. Steven Mackey ◽  
Niloufer Mackey ◽  
Françoise Tisseur
Keyword(s):  
Scalar Product ◽  
Product Spaces
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