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The Gau–Wu number for 4 × 4 and select arrowhead matrices

The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e.,...

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Bibliographic Details
Published in:Linear algebra and its applications 2022-07, Vol.644, p.192-218
Main Authors: Camenga, Kristin A., Rault, Patrick X., Spitkovsky, Ilya M., Yates, Rebekah B. Johnson
Format: Article
Language:English
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Summary:The notion of dichotomous matrices is introduced as a natural generalization of essentially Hermitian matrices. A criterion for arrowhead matrices to be dichotomous is established, along with necessary and sufficient conditions for such matrices to be unitarily irreducible. The Gau–Wu number (i.e., the maximal number k(A) of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range of A) is computed for a class of arrowhead matrices A of arbitrary size, including dichotomous ones. These results are then used to completely classify all 4×4 matrices according to the values of their Gau–Wu numbers.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2022.02.026