Sparse orthogonal matrices.
type: Item
instance of: scholarly article
title: Sparse orthogonal matrices.
zbMATH Open document ID: 1035.15022
DOI: 10.1016/S0024-3795(03)00389-6
author: Gi-Sang Cheon
author: Suk-Geun Hwang
author: Bryan L. Shader
author: Seok-Zun Song
author: Seog-Hoon Rim
published in: Linear Algebra and its Applications
publication date: 2003-11-19T00:00:00Z
review text: The sparsity of orthogonal matrices which have both a column and a row of nonzero is studied. In Section 2, the authors describe a rich family \(n\) by \(n\) orthogonal matrices, namely, those that are the product of \(n-1\) Givens rotations. They show that this family contains a sparsest fully indecomposable orthogonal matrix with a full row. They also show that the smallest number of nonzero entries in a matrix which is in this family and which has both a full row and a full column is \((n^2+ 3n-2)/2\). In Section 3, for \(n\geq 6\) they construct \(n\) by \(n\) orthogonal matrices with both a full row and column that have \[ g(n):= \begin{cases} (2k+2)n- 7\cdot 2^{k-1}+ 3\quad &\text{if }2^k\leq n\leq 2^k+ 2^{k-1},\\ (2k+ 3)n- 10\cdot 2^{k-1}+ 3\quad &\text{if }2^k+ 2^{k-1}< n< 2^{k+1}\end{cases} \] nonzero entries, which is significantly smaller than the sparsity, \((n^2+ 3n- 2)/2\), for such an orthogonal matrix that is the product of \(n-1\) Givens rotations.
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 2005982
zbMATH Keywords: sparse orthogonal matrix
zbMATH Keywords: Hessenberg matrix
zbMATH Keywords: Givens rotations
Wikidata QID: Q57702745
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
cites work: Q3218140
cites work: Q4281318
cites work: The possible numbers of zeros in a orthogonal matrix
cites work: How sparse can a matrix with orthogonal rows be?
cites work: Sparse orthogonal matrices and the Haar wavelet
cites work: Sparsity of orthogonal matrices with restrictions
cites work: Q4312077
cites work: Sign patterns occurring in orthogonal matrices
cites work: A simple proof of Fiedler's conjecture concerning orthogonal matrices
cites work: Sign pattern matrices that allow orthogonality
description: scientific article; zbMATH DE number 2005982