Birkhoff-James approximate orthogonality sets and numerical ranges
type: Item
instance of: scholarly article
title: Birkhoff-James approximate orthogonality sets and numerical ranges
zbMATH Open document ID: 1215.15024
author: Christos Chorianopoulos
author: Panayiotis J. Psarrakos
published in: Linear Algebra and its Applications
publication date: 2011-03-17T00:00:00Z
review text: For any \(n\)-by-\(m\) matrices \(A\) and \(B\) with \(B\neq 0\), any matrix norm \(\|.\|\) and any \(\varepsilon\), \(0\leq\varepsilon< 1\), the Birkhoff-James \(\varepsilon\)-orthogonality set of \(A\) with respect to \(B\) is defined as \[ F^\varepsilon_{\|.\|}(A; B)= \{u\in\mathbb{C}:\| A-\lambda B\|\geq (1-\varepsilon^2)^{1/2}\| B\|\,|u-\lambda|\text{ for all }\lambda\text{ in }\mathbb{C}\}. \] This is a generalization of the classical numerical range (when \(n= m\), \(\|.\|\) is the spectral norm, \(\varepsilon= 0\) and \(B= I_n\)) and the \(q\)-numerical range (when \(n= m\), \(\|.\|\) and \(B\) as above, and \(\varepsilon= q\)) of square matrices. This approach in various forms has also been adopted by other people in recent years. The above-defined sets are easily seen to be nonempty, compact and convex. Other basic properties compatible with the classical case are proven in Section 2 of this paper. Then in Section 3, the Birkhoff-James \(\varepsilon\)-orthogonality set of a matrix polynomial \[ p(z)= A_\ell z^\ell+ A_{\ell-1} z^{\ell-1}+\cdots+ A_1 z+ A_0, \] where the \(A_j\)'s are \(n\)-by-\(m\) matrices with \(A_\ell\neq 0\), with respect to \(B\neq 0\) is also defined by \[ W^\varepsilon_{\|.\|}(p(z); B)= \{u\in\mathbb{C}:\| p(u)-\lambda B\|\geq (1-\varepsilon^2)^{1/2}\| B\|\, |\lambda|\text{ for all }\lambda\text{ in }\mathbb{C}\}, \] for any matrix norm \(\|.\|\) and any \(\varepsilon\), \(0\leq\varepsilon< 1\), which generalizes the standard numerical range and \(q\)-numerical range of \(p(z)\). Section 4 then treats properties of the boundary points of these \(\varepsilon\)-orthogonality sets. In particular, it is shown that if \(u_0\) is in \(\partial W^\varepsilon_{\|.\|}(p(z); B)\), then \(0\) is in \(\partial F^\varepsilon_{\|.\|}(p(u_0); B)\). The paper concludes in Section 5 with some simplified expressions for both \(\varepsilon\)-orthogonality sets when the matrix norm \(\|.\|\) is induced by an inner product of matrices.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 5867199
zbMATH Keywords: Birkhoff-James orthogonality
zbMATH Keywords: numerical range
zbMATH Keywords: matrix polynomial
zbMATH Keywords: eigenvalue
zbMATH Keywords: rectangular matrix
zbMATH Keywords: boundary
zbMATH Keywords: spectral norm
zbMATH Keywords: \(q\)-numerical range
zbMATH Keywords: inner product of matrices
MaRDI profile type: Publication
full work available at URL: j.laa.2010.12.008
OpenAlex ID: W2025845594
cites work: Investigating the numerical range and q-numerical range of non square matrices
cites work: Orthogonality in linear metric spaces
cites work: Q5608364
cites work: Q5678038
cites work: The boundary of theq-numerical range of a reducible matrix
cites work: On theq-numerical range of matrices and matrix polynomials
cites work: Q5699867
cites work: A definition of numerical range of rectangular matrices
cites work: Q4280124
cites work: Q3939931
cites work: Q4040931
cites work: Orthogonality and Linear Functionals in Normed Linear Spaces
cites work: Q4110164
cites work: Normal and seminormal eigenvalues of matrix functions
cites work: Inertia characteristics of self-adjoint matrix polynomials
cites work: A generalization of the numerical range of a matrix
cites work: A generalized numerical range: the range of a constrained sesquilinear form
cites work: Some Results on the q-Numerical
cites work: Numerical Range of Matrix Polynomials
cites work: Constrained extrema of bilinear functionals
cites work: Q3994769
cites work: The boundary of the numerical range of matrix polynomials
cites work: On the connectedness of numerical range of matrix polynomials
cites work: On the shape of numerical range of matrix polynomials
cites work: On the relation between the numerical range and the joint numerical range of matrix polynomials
cites work: Growth conitions and the numerical range in a Banach algebra
DOI: 10.1016/J.LAA.2010.12.008
description: scientific article; zbMATH DE number 5867199