### 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} : {x_{1}, . . . , x_{k}} is an orthonormal set in ℂ^{n}}. It is known that {tr A/n} = W_{n}(A) ⊆ W_{n-1}(A) ⊆ ⋯ ⊆ W_{1}(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 | |

State | Published - Jan 1 1998 |

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### ASJC Scopus subject areas

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

### Cite this

*Linear Algebra and Its Applications*,

*270*(1-3), 323-349. https://doi.org/10.1016/S0024-3795(97)00251-6

**Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils.** / Li, Chi Kwong; Spitkovsky, Ilya; Shukla, Sudheer.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 270, no. 1-3, pp. 323-349. https://doi.org/10.1016/S0024-3795(97)00251-6

}

TY - JOUR

T1 - Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

AU - Li, Chi Kwong

AU - Spitkovsky, Ilya

AU - Shukla, Sudheer

PY - 1998/1/1

Y1 - 1998/1/1

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)Σj k=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)Σj k=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.

UR - http://www.scopus.com/inward/record.url?scp=0042027212&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042027212&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(97)00251-6

DO - 10.1016/S0024-3795(97)00251-6

M3 - Article

VL - 270

SP - 323

EP - 349

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -