TY - JOUR

T1 - Numerical radii for tensor products of matrices

AU - Gau, Hwa Long

AU - Wang, Kuo Zhong

AU - Wu, Pei Yuan

N1 - Publisher Copyright:
© 2013 Taylor & Francis.

PY - 2015/10/3

Y1 - 2015/10/3

N2 - Abstract: For (Formula presented.) -by- (Formula presented.) and (Formula presented.) -by- (Formula presented.) complex matrices (Formula presented.) and (Formula presented.) , it is known that the inequality (Formula presented.) holds, where (Formula presented.) and (Formula presented.) denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if (Formula presented.) and (Formula presented.) , then one of the following two conditions holds: (i) (Formula presented.) has a unitary part, and (ii) (Formula presented.) is completely nonunitary and the numerical range (Formula presented.) of (Formula presented.) is a circular disc centered at the origin, (2) if (Formula presented.) for some (Formula presented.) , (Formula presented.) , then (Formula presented.) , and, moreover, the equality holds if and only if (Formula presented.) is unitarily similar to the direct sum of the (Formula presented.) -by- (Formula presented.) Jordan block (Formula presented.) and a matrix (Formula presented.) with (Formula presented.) , and (3) if (Formula presented.) is a nonnegative matrix with its real part (permutationally) irreducible, then (Formula presented.) , if and only if either (Formula presented.) or (Formula presented.) and (Formula presented.) is permutationally similar to a block-shift matrix (Formula presented.) with (Formula presented.) , where (Formula presented.) and (Formula presented.).

AB - Abstract: For (Formula presented.) -by- (Formula presented.) and (Formula presented.) -by- (Formula presented.) complex matrices (Formula presented.) and (Formula presented.) , it is known that the inequality (Formula presented.) holds, where (Formula presented.) and (Formula presented.) denote, respectively, the numerical radius and the operator norm of a matrix. In this paper, we consider when this becomes an equality. We show that (1) if (Formula presented.) and (Formula presented.) , then one of the following two conditions holds: (i) (Formula presented.) has a unitary part, and (ii) (Formula presented.) is completely nonunitary and the numerical range (Formula presented.) of (Formula presented.) is a circular disc centered at the origin, (2) if (Formula presented.) for some (Formula presented.) , (Formula presented.) , then (Formula presented.) , and, moreover, the equality holds if and only if (Formula presented.) is unitarily similar to the direct sum of the (Formula presented.) -by- (Formula presented.) Jordan block (Formula presented.) and a matrix (Formula presented.) with (Formula presented.) , and (3) if (Formula presented.) is a nonnegative matrix with its real part (permutationally) irreducible, then (Formula presented.) , if and only if either (Formula presented.) or (Formula presented.) and (Formula presented.) is permutationally similar to a block-shift matrix (Formula presented.) with (Formula presented.) , where (Formula presented.) and (Formula presented.).

KW - Sn-matrix

KW - nonnegative matrix

KW - numerical radius

KW - numerical range

KW - tensor product

UR - http://www.scopus.com/inward/record.url?scp=84938971054&partnerID=8YFLogxK

U2 - 10.1080/03081087.2013.839669

DO - 10.1080/03081087.2013.839669

M3 - 期刊論文

AN - SCOPUS:84938971054

SN - 0308-1087

VL - 63

SP - 1916

EP - 1936

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

IS - 10

ER -