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

✍ Scribed by Chi-Kwong Li; Ilya Spitkovsky; Sudheer Shukla

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 997 KB
• Volume: 270
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(97)00251-6

✦ Synopsis

Let M n be the algebra of all n × n complex matrices. For 1 ~< k ~< n, the kth numerical range of A 6 M, is defined by Wk(A) = {(i/k)E~=,x*~Axj: {x t ..... x k} is an orthonormal set in C"}. It is known that {tr A/n} = W,(A) ~ W n_ 1(A) c_ ... c_ WI(A). We study the condition on A under which W~(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.