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NUMERICAL RANGE AND POSITIVE BLOCK MATRICES

Part of: General theory of linear operators Basic linear algebra

Published online by Cambridge University Press:  11 June 2020

JEAN-CHRISTOPHE BOURIN  and
EUN-YOUNG LEE
JEAN-CHRISTOPHE BOURIN
Affiliation:
Laboratoire de Mathématiques de Besançon, UMR 6623, CNRS, Université de Bourgogne Franche-Comté, Besançon, France email jcbourin@univ-fcomte.fr
EUN-YOUNG LEE*
Affiliation:
Department of Mathematics, KNU-Center for Nonlinear Dynamics, Kyungpook National University, Daegu 702-701, Korea email eylee89@knu.ac.kr
*
✉eylee89@knu.ac.kr
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Abstract

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate,

$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$
between the diameters of the numerical ranges for the full matrix and its partial trace.

Keywords

numerical rangepartitioned matricesnorm inequalities

MSC classification

Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory
Secondary: 15A42: Inequalities involving eigenvalues and eigenvectors 47A12: Numerical range, numerical radius 47A30: Norms (inequalities, more than one norm, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 69 - 77
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The first author was funded by the ANR Project No. ANR-19-CE40-0002 and by the French Investissements d’Avenir program, project ISITE-BFC (contract ANR-15-IDEX-03). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07043682).

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