Equality of higher-rank numerical ranges of matrices

Chi Tung Chang; Hwa Long Gau; Kuo Zhong Wang

doi:10.1080/03081087.2013.811500

Equality of higher-rank numerical ranges of matrices

Chi Tung Chang, Hwa Long Gau, Kuo Zhong Wang

Department of Mathematics

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

6 Scopus citations

Abstract

Let Λ_k (A) denote the rank-k numerical range of an n-by-n complex matrix A. We give a characterization for Λ_k1 (A) = Λ_k2 (A), where 1 ≤_k1 ≤_k2 ≤ n, via the compressions and the principal submatrices of A. As an application, the matrix A satisfying W(A) = Λ_k (A), where W(A) is the classical numerical range of A and 1 ≤ k ≤ n, is under consideration. We show that if W(A) = Λ_k (A) for some k > n/3, then A is unitarily similar to B ⊗ B ⊗ ... ⊗ B⊗C, where B is a 2-by-2 matrix, C is a (3n - 6k)-by-(3n - 6k) matrix and W(A) = W(B) = W(C) Λ_n-2k (C).

Original language	English
Pages (from-to)	626-638
Number of pages	13
Journal	Linear and Multilinear Algebra
Volume	62
Issue number	5
DOIs	https://doi.org/10.1080/03081087.2013.811500
State	Published - May 2014

Keywords

compression
higher-rank numerical range
numerical range
principal submatrix

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title = "Equality of higher-rank numerical ranges of matrices",

abstract = "Let Λk (A) denote the rank-k numerical range of an n-by-n complex matrix A. We give a characterization for Λk1 (A) = Λk2 (A), where 1 ≤k1 ≤k2 ≤ n, via the compressions and the principal submatrices of A. As an application, the matrix A satisfying W(A) = Λk (A), where W(A) is the classical numerical range of A and 1 ≤ k ≤ n, is under consideration. We show that if W(A) = Λk (A) for some k > n/3, then A is unitarily similar to B ⊗ B ⊗ ... ⊗ B⊗C, where B is a 2-by-2 matrix, C is a (3n - 6k)-by-(3n - 6k) matrix and W(A) = W(B) = W(C) Λn-2k (C).",

keywords = "compression, higher-rank numerical range, numerical range, principal submatrix",

author = "Chang, {Chi Tung} and Gau, {Hwa Long} and Wang, {Kuo Zhong}",

note = "Funding Information: We appreciate the advice from Professor Pei Yuan Wu. He pointed out that there exists an 6-by-6 unitarily irreducible matrix A such that W(A) = Λ2(A) and therefore, the number n/3 in Theorem 2.9 (c) is best possible. We also thank the referee for his/her comments, which improved both the statement and the proof of Theorem 2.2. The Research was suppported by the National Science Council of the Republic of China under the projects NSC 101-2115-M-035-006, NSC 101-2115-M-008-006 and NSC 101-2115-M-009-001, respectively.",

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AU - Chang, Chi Tung

AU - Gau, Hwa Long

AU - Wang, Kuo Zhong

N1 - Funding Information: We appreciate the advice from Professor Pei Yuan Wu. He pointed out that there exists an 6-by-6 unitarily irreducible matrix A such that W(A) = Λ2(A) and therefore, the number n/3 in Theorem 2.9 (c) is best possible. We also thank the referee for his/her comments, which improved both the statement and the proof of Theorem 2.2. The Research was suppported by the National Science Council of the Republic of China under the projects NSC 101-2115-M-035-006, NSC 101-2115-M-008-006 and NSC 101-2115-M-009-001, respectively.

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N2 - Let Λk (A) denote the rank-k numerical range of an n-by-n complex matrix A. We give a characterization for Λk1 (A) = Λk2 (A), where 1 ≤k1 ≤k2 ≤ n, via the compressions and the principal submatrices of A. As an application, the matrix A satisfying W(A) = Λk (A), where W(A) is the classical numerical range of A and 1 ≤ k ≤ n, is under consideration. We show that if W(A) = Λk (A) for some k > n/3, then A is unitarily similar to B ⊗ B ⊗ ... ⊗ B⊗C, where B is a 2-by-2 matrix, C is a (3n - 6k)-by-(3n - 6k) matrix and W(A) = W(B) = W(C) Λn-2k (C).

AB - Let Λk (A) denote the rank-k numerical range of an n-by-n complex matrix A. We give a characterization for Λk1 (A) = Λk2 (A), where 1 ≤k1 ≤k2 ≤ n, via the compressions and the principal submatrices of A. As an application, the matrix A satisfying W(A) = Λk (A), where W(A) is the classical numerical range of A and 1 ≤ k ≤ n, is under consideration. We show that if W(A) = Λk (A) for some k > n/3, then A is unitarily similar to B ⊗ B ⊗ ... ⊗ B⊗C, where B is a 2-by-2 matrix, C is a (3n - 6k)-by-(3n - 6k) matrix and W(A) = W(B) = W(C) Λn-2k (C).

KW - compression

KW - higher-rank numerical range

KW - numerical range

KW - principal submatrix

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DO - 10.1080/03081087.2013.811500

M3 - 期刊論文

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JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

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ER -

Equality of higher-rank numerical ranges of matrices

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