Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Chi Kwong Li; Ilya Spitkovsky; Sudheer Shukla

doi:10.1016/S0024-3795(97)00251-6

Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Chi Kwong Li, Ilya Spitkovsky, Sudheer Shukla

Mathematics

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

Abstract

Let M_n be the algebra of all n × n complex matrices. For 1 ≤ k ≤ n, the kth numerical range of A ∈ M_n is defined by W_k(A) = {(1/k)Σ_j^k=1 x_j*Ax_j : {x₁, . . . , x_k} is an orthonormal set in ℂⁿ}. It is known that {tr A/n} = W_n(A) ⊆ W_n-1(A) ⊆ ⋯ ⊆ W₁(A). We study the condition on A under which W_m(A) = W_k(A) for some given 1 ≤ m < k ≤ n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.

Original language	English (US)
Pages (from-to)	323-349
Number of pages	27
Journal	Linear Algebra and Its Applications
Volume	270
Issue number	1-3
DOIs	https://doi.org/10.1016/S0024-3795(97)00251-6
State	Published - Feb 1998

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.1016/S0024-3795(97)00251-6

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title = "Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils",

abstract = "Let Mn be the algebra of all n × n complex matrices. For 1 ≤ k ≤ n, the kth numerical range of A ∈ Mn is defined by Wk(A) = {(1/k)Σjk=1 xj*Axj : {x1, . . . , xk} is an orthonormal set in ℂn}. It is known that {tr A/n} = Wn(A) ⊆ Wn-1(A) ⊆ ⋯ ⊆ W1(A). We study the condition on A under which Wm(A) = Wk(A) for some given 1 ≤ m < k ≤ n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.",

author = "Li, {Chi Kwong} and Ilya Spitkovsky and Sudheer Shukla",

note = "Funding Information: * Partially supported by a NATO grant. E-maih ckl i@math, win. edu. Funding Information: t Part~s\]lsyu pported by NSF grant DMS 9401848. E-mail: ilya@math .win.e du. SResearch was done in the summer of 1995 at the College of William and Mary supported by NSF Research Experience for Undergraduates grant 9300395. E-mail: sshukla@ wam. umd, edu.",

year = "1998",

month = feb,

doi = "10.1016/S0024-3795(97)00251-6",

language = "English (US)",

volume = "270",

pages = "323--349",

journal = "Linear Algebra and Its Applications",

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AU - Li, Chi Kwong

AU - Spitkovsky, Ilya

AU - Shukla, Sudheer

N1 - Funding Information: * Partially supported by a NATO grant. E-maih ckl i@math, win. edu. Funding Information: t Part~s\]lsyu pported by NSF grant DMS 9401848. E-mail: ilya@math .win.e du. SResearch was done in the summer of 1995 at the College of William and Mary supported by NSF Research Experience for Undergraduates grant 9300395. E-mail: sshukla@ wam. umd, edu.

PY - 1998/2

Y1 - 1998/2

N2 - Let Mn be the algebra of all n × n complex matrices. For 1 ≤ k ≤ n, the kth numerical range of A ∈ Mn is defined by Wk(A) = {(1/k)Σjk=1 xj*Axj : {x1, . . . , xk} is an orthonormal set in ℂn}. It is known that {tr A/n} = Wn(A) ⊆ Wn-1(A) ⊆ ⋯ ⊆ W1(A). We study the condition on A under which Wm(A) = Wk(A) for some given 1 ≤ m < k ≤ n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.

AB - Let Mn be the algebra of all n × n complex matrices. For 1 ≤ k ≤ n, the kth numerical range of A ∈ Mn is defined by Wk(A) = {(1/k)Σjk=1 xj*Axj : {x1, . . . , xk} is an orthonormal set in ℂn}. It is known that {tr A/n} = Wn(A) ⊆ Wn-1(A) ⊆ ⋯ ⊆ W1(A). We study the condition on A under which Wm(A) = Wk(A) for some given 1 ≤ m < k ≤ n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.

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Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

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