On the C-determinantal range for special classes of matrices

Alexander Guterman; Rute Lemos; Graça Soares

doi:10.1016/j.amc.2015.11.042

On the C-determinantal range for special classes of matrices

Alexander Guterman, Rute Lemos, Graça Soares

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

Abstract

Let A and C be square complex matrices of size n, the C-determinantal range of A is the subset of the complex plane {det(A-UC^U∗):U^U∗=_In}. If A, C are both Hermitian matrices, then by a result of Fiedler (1971) [11] this set is a real line segment. In our paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1 (a) in Li, (1994) [20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, (see Li et al. (2008) [21], Theorem 3.3).

Original language	English
Pages (from-to)	86-94
Number of pages	9
Journal	Applied Mathematics and Computation
Volume	275
DOIs	https://doi.org/10.1016/j.amc.2015.11.042
State	Published - 15 Feb 2016
Externally published	Yes

Keywords

C-determinantal range
C-numerical range
Marcus-Oliveira conjecture
Real sets
σ-points

All Science Journal Classification (ASJC) codes

Computational Mathematics
Applied Mathematics

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10.1016/j.amc.2015.11.042

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title = "On the C-determinantal range for special classes of matrices",

abstract = "Let A and C be square complex matrices of size n, the C-determinantal range of A is the subset of the complex plane \{det(A-UCU∗):UU∗=In\}. If A, C are both Hermitian matrices, then by a result of Fiedler (1971) [11] this set is a real line segment. In our paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1 (a) in Li, (1994) [20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, (see Li et al. (2008) [21], Theorem 3.3).",

keywords = "C-determinantal range, C-numerical range, Marcus-Oliveira conjecture, Real sets, σ-points",

author = "Alexander Guterman and Rute Lemos and Gra{\c c}a Soares",

year = "2016",

month = feb,

day = "15",

doi = "10.1016/j.amc.2015.11.042",

language = "الإنجليزيّة",

volume = "275",

pages = "86--94",

journal = "Applied Mathematics and Computation",

issn = "0096-3003",

publisher = "Elsevier Inc.",

}

TY - JOUR

T1 - On the C-determinantal range for special classes of matrices

AU - Guterman, Alexander

AU - Lemos, Rute

AU - Soares, Graça

PY - 2016/2/15

Y1 - 2016/2/15

N2 - Let A and C be square complex matrices of size n, the C-determinantal range of A is the subset of the complex plane {det(A-UCU∗):UU∗=In}. If A, C are both Hermitian matrices, then by a result of Fiedler (1971) [11] this set is a real line segment. In our paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1 (a) in Li, (1994) [20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, (see Li et al. (2008) [21], Theorem 3.3).

AB - Let A and C be square complex matrices of size n, the C-determinantal range of A is the subset of the complex plane {det(A-UCU∗):UU∗=In}. If A, C are both Hermitian matrices, then by a result of Fiedler (1971) [11] this set is a real line segment. In our paper we study this set for the case when C is a Hermitian matrix. Our purpose is to revisit and improve two well-known results on this topic. The first result is due to Li concerning the C-numerical range of a Hermitian matrix, see Condition 5.1 (a) in Li, (1994) [20]. The second one is due to C.-K. Li, Y.-T. Poon and N.-S. Sze about necessary and sufficient conditions for the C-determinantal range of A to be a subset of the line, (see Li et al. (2008) [21], Theorem 3.3).

KW - C-determinantal range

KW - C-numerical range

KW - Marcus-Oliveira conjecture

KW - Real sets

KW - σ-points

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U2 - 10.1016/j.amc.2015.11.042

DO - 10.1016/j.amc.2015.11.042

M3 - مقالة

SN - 0096-3003

VL - 275

SP - 86

EP - 94

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

ER -

On the C-determinantal range for special classes of matrices

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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