کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4598645 1631094 2016 20 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Extremality of numerical radii of matrix products
ترجمه فارسی عنوان
افراطی کردن شعاع عددی محصولات ماتریسی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
چکیده انگلیسی

For two n-by-n matrices A and B  , it was known before that their numerical radii satisfy the inequality w(AB)≤4w(A)w(B)w(AB)≤4w(A)w(B), and the equality is attained by the 2-by-2 matrices A=[0100] and B=[0010]. Moreover, the constant “4” here can be reduced to “2” if A and B   commute, and the corresponding equality is attained by A=I2⊗[0100] and B=[0100]⊗I2. In this paper, we give a complete characterization of A and B   for which the equality holds in each case. More precisely, it is shown that w(AB)=4w(A)w(B)w(AB)=4w(A)w(B) (resp., w(AB)=2w(A)w(B)w(AB)=2w(A)w(B) for commuting A and B) if and only if either A or B is the zero matrix, or A and B   are simultaneously unitarily similar to matrices of the form [0a00]⊕A′ and [00b0]⊕B′ (resp.,⊕A′and⊕B′) with w(A′)≤|a|/2w(A′)≤|a|/2 and w(B′)≤|b|/2w(B′)≤|b|/2. An analogous characterization for the extremal equality for tensor products is also proven. For doubly commuting matrices, we use their unitary similarity model to obtain the corresponding result. For commuting 2-by-2 matrices A and B  , we show that w(AB)=w(A)w(B)w(AB)=w(A)w(B) if and only if either A or B is a scalar matrix, or A and B   are simultaneously unitarily similar to [a100a2] and [b100b2] with |a1|≥|a2||a1|≥|a2| and |b1|≥|b2||b1|≥|b2|.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 501, 15 July 2016, Pages 17–36
نویسندگان
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