Exterior numerical radii and antiinvariant subspaces

doi:10.1080/03081089808818560

機構典藏 > College of Sciences > Department of Applied Mathematics and Data Science > Journal Article > Item 987654321/41412

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Title:	Exterior numerical radii and antiinvariant subspaces
Authors:	譚必信;Tam, Bit-shun
Contributors:	淡江大學數學學系
Keywords:	mth A-exterior numerical radius;Marcus and Sandy conjecture;matching of a multi-set;antiinvariant subspace;gap between subspaces
Date:	1998-07-01
Issue Date:	2010-01-28 07:30:21 (UTC+8)
Publisher:	Taylor & Francis
Abstract:	Let m be a positive integer less than n, and let A B be n×n complex matrices. The mth A-exterior numerical radius of B is defined as where Cm( A) is the mth compound matrix of A and Un is the group of n×n unitary matrices. Marcus and Sandy conjectured that a (necessary and) sufficient condition for to imply that the rank of B is less than m is that A is nonscalar and counter-example to the conjecture was given in [10]. In this paper, using geometric arguments, we show that the conjecture is true in the special case when A is of rank m. Then we prove that in order that for each B of rank m it is necessary and sufficient that rank A≤m and rank (A—λI)≤min{m, n—m) for all complex numbers λ. We also find other equivalent conditions one of which is that, for any pair of linear subspaces X 9 Y of Cn with dim X= m and dim Y =n— m, there exists UεUn such that Cn = AU(X)⊗U(Y). To establish the equivalence of the above conditions, we give a structure theory of the anti-invariance of matrices, which relies on a combinatorial method and the use of gap between subspaces. As by-products, we obtain two interesting results about matchings of a multi-set. The cases when Unis replaced by GL(n, C) or the underlying field is the set of real numbers are also examined carefully. Some open problems are posed at the end.
Relation:	Linear and Multilinear Algebra 44(3), pp.195-245
DOI:	10.1080/03081089808818560
Appears in Collections:	[Department of Applied Mathematics and Data Science] Journal Article

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