On the maximal numerical range of some matrices

Ali N. Hamed; Ilya M. Spitkovsky

doi:10.13001/1081-3810.3774

On the maximal numerical range of some matrices

Ali N. Hamed, Ilya M. Spitkovsky

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The maximal numerical range W₀(A) of a matrix A is the (regular) numerical range W (B) of its compression B onto the eigenspace L of A*A corresponding to its maximal eigenvalue. So, always W₀(A) ⊆ W (A). Conditions under which W₀(A) has a non-empty intersection with the boundary of W (A) are established, in particular, when W₀(A) = W (A). The set W₀(A) is also described explicitly for matrices unitarily similar to direct sums of 2-by-2 blocks, and some insight into the behavior of W₀(A) is provided when L has codimension one.

Original language	English (US)
Article number	21
Pages (from-to)	288-303
Number of pages	16
Journal	Electronic Journal of Linear Algebra
Volume	34
DOIs	https://doi.org/10.13001/1081-3810.3774
State	Published - 2018

Keywords

Maximal numerical range
Normaloid matrices
Numerical range

ASJC Scopus subject areas

Algebra and Number Theory

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title = "On the maximal numerical range of some matrices",

abstract = "The maximal numerical range W0(A) of a matrix A is the (regular) numerical range W (B) of its compression B onto the eigenspace L of A*A corresponding to its maximal eigenvalue. So, always W0(A) ⊆ W (A). Conditions under which W0(A) has a non-empty intersection with the boundary of W (A) are established, in particular, when W0(A) = W (A). The set W0(A) is also described explicitly for matrices unitarily similar to direct sums of 2-by-2 blocks, and some insight into the behavior of W0(A) is provided when L has codimension one.",

keywords = "Maximal numerical range, Normaloid matrices, Numerical range",

author = "Hamed, {Ali N.} and Spitkovsky, {Ilya M.}",

note = "Funding Information: ‡Division of Science and Mathematics, New York University Abu Dhabi (NYUAD), Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates (ims2@nyu.edu, imspitkovsky@gmail.com). Supported in part by Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi. Publisher Copyright: {\textcopyright} 2018, International Linear Algebra Society. All rights reserved.",

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doi = "10.13001/1081-3810.3774",

language = "English (US)",

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AU - Hamed, Ali N.

AU - Spitkovsky, Ilya M.

N1 - Funding Information: ‡Division of Science and Mathematics, New York University Abu Dhabi (NYUAD), Saadiyat Island, P.O. Box 129188, Abu Dhabi, United Arab Emirates (ims2@nyu.edu, imspitkovsky@gmail.com). Supported in part by Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi. Publisher Copyright: © 2018, International Linear Algebra Society. All rights reserved.

PY - 2018

Y1 - 2018

N2 - The maximal numerical range W0(A) of a matrix A is the (regular) numerical range W (B) of its compression B onto the eigenspace L of A*A corresponding to its maximal eigenvalue. So, always W0(A) ⊆ W (A). Conditions under which W0(A) has a non-empty intersection with the boundary of W (A) are established, in particular, when W0(A) = W (A). The set W0(A) is also described explicitly for matrices unitarily similar to direct sums of 2-by-2 blocks, and some insight into the behavior of W0(A) is provided when L has codimension one.

AB - The maximal numerical range W0(A) of a matrix A is the (regular) numerical range W (B) of its compression B onto the eigenspace L of A*A corresponding to its maximal eigenvalue. So, always W0(A) ⊆ W (A). Conditions under which W0(A) has a non-empty intersection with the boundary of W (A) are established, in particular, when W0(A) = W (A). The set W0(A) is also described explicitly for matrices unitarily similar to direct sums of 2-by-2 blocks, and some insight into the behavior of W0(A) is provided when L has codimension one.

KW - Maximal numerical range

KW - Normaloid matrices

KW - Numerical range

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On the maximal numerical range of some matrices

Abstract

Keywords

ASJC Scopus subject areas

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