Premium
Triangular truncation and its extremal matrices
Author(s) -
Zhou Weiqi
Publication year - 2016
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2045
Subject(s) - mathematics , toeplitz matrix , operator norm , matrix norm , triangular matrix , norm (philosophy) , truncation (statistics) , logarithm , linear map , condition number , operator (biology) , matrix (chemical analysis) , pure mathematics , mathematical analysis , combinatorics , eigenvalues and eigenvectors , operator theory , invertible matrix , statistics , biochemistry , physics , chemistry , materials science , repressor , quantum mechanics , composite material , political science , transcription factor , law , gene
Summary The triangular truncation operator is a linear transformation that maps a given matrix to its strictly lower triangular part. The operator norm (with respect to the matrix spectral norm) of the triangular truncation is known to have logarithmic dependence on the dimension, and such dependence is usually illustrated by a specific Toeplitz matrix. However, the precise value of this operator norm as well as on which matrices can it be attained is still unclear. In this article, we describe a simple way of constructing matrices whose strictly lower triangular part has logarithmically larger spectral norm. The construction also leads to a sharp estimate that is very close to the actual operator norm of the triangular truncation. This research is directly motivated by our studies on the convergence theory of the Kaczmarz type method (or equivalently, the Gauß‐Seidel type method), the corresponding application of which is also included. Copyright © 2016 John Wiley & Sons, Ltd.