Numerical ranges of kms matrices

Hwa Long Gau; Pei Yuan Wu

Numerical ranges of kms matrices

Hwa Long Gau, Pei Yuan Wu

Department of Mathematics

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

2 Scopus citations

Abstract

We prove the following properties of the numerical range of a KMS matrix J_n(a): (1) W(J_n(a)) is a circular disc if and only if n = 2 and a ≠ 0, (2) its boundary ?W(J_n(a)) contains a line segment if and only if n ≥ 3 and |a| = 1, and (3) the intersection of the boundaries ?W(J_n(a)) and ∂W(J_n(a)[j]) is either the singleton {min s(Re J_n(a))} if n is odd, j = (n + 1)/2 and |a| > 1, or the empty set Ø if otherwise, where, for any n-by-n matrix A, A[j] denotes its jth principal submatrix obtained by deleting its jth row and jth column (1 ≤ j ≤ n), ReA its real part (A + A*)/2, and s(A) its spectrum.

Original language	English
Pages (from-to)	583-610
Number of pages	28
Journal	Acta Scientiarum Mathematicarum
Volume	79
Issue number	3-4
State	Published - 2013

Keywords

KMS matrix
Numerical range
S -matrix
S-matrix

Cite this

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abstract = "We prove the following properties of the numerical range of a KMS matrix Jn(a): (1) W(Jn(a)) is a circular disc if and only if n = 2 and a ≠ 0, (2) its boundary ?W(Jn(a)) contains a line segment if and only if n ≥ 3 and |a| = 1, and (3) the intersection of the boundaries ?W(Jn(a)) and ∂W(Jn(a)[j]) is either the singleton {min s(Re Jn(a))} if n is odd, j = (n + 1)/2 and |a| > 1, or the empty set {\O} if otherwise, where, for any n-by-n matrix A, A[j] denotes its jth principal submatrix obtained by deleting its jth row and jth column (1 ≤ j ≤ n), ReA its real part (A + A*)/2, and s(A) its spectrum.",

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AU - Wu, Pei Yuan

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N2 - We prove the following properties of the numerical range of a KMS matrix Jn(a): (1) W(Jn(a)) is a circular disc if and only if n = 2 and a ≠ 0, (2) its boundary ?W(Jn(a)) contains a line segment if and only if n ≥ 3 and |a| = 1, and (3) the intersection of the boundaries ?W(Jn(a)) and ∂W(Jn(a)[j]) is either the singleton {min s(Re Jn(a))} if n is odd, j = (n + 1)/2 and |a| > 1, or the empty set Ø if otherwise, where, for any n-by-n matrix A, A[j] denotes its jth principal submatrix obtained by deleting its jth row and jth column (1 ≤ j ≤ n), ReA its real part (A + A*)/2, and s(A) its spectrum.

AB - We prove the following properties of the numerical range of a KMS matrix Jn(a): (1) W(Jn(a)) is a circular disc if and only if n = 2 and a ≠ 0, (2) its boundary ?W(Jn(a)) contains a line segment if and only if n ≥ 3 and |a| = 1, and (3) the intersection of the boundaries ?W(Jn(a)) and ∂W(Jn(a)[j]) is either the singleton {min s(Re Jn(a))} if n is odd, j = (n + 1)/2 and |a| > 1, or the empty set Ø if otherwise, where, for any n-by-n matrix A, A[j] denotes its jth principal submatrix obtained by deleting its jth row and jth column (1 ≤ j ≤ n), ReA its real part (A + A*)/2, and s(A) its spectrum.

KW - KMS matrix

KW - Numerical range

KW - S -matrix

KW - S-matrix

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M3 - 期刊論文

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JF - Acta Scientiarum Mathematicarum

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Numerical ranges of kms matrices

Abstract

Keywords

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