TY - JOUR

T1 - Singularities of base polynomials and Gau–Wu numbers

AU - Camenga, Kristin A.

AU - Deaett, Louis

AU - Rault, Patrick X.

AU - Sendova, Tsvetanka

AU - Spitkovsky, Ilya M.

AU - Yates, Rebekah B.Johnson

N1 - Funding Information:
The work on this paper was prompted by the authors' discussions during a SQuaRE workshop in May 2013 and continued during a REUF continuation workshop in August 2017, both at the American Institute of Mathematics (AIM) and supported by the National Science Foundation grant number DMS-1620073. AIM also provided financial support for an additional meeting finalizing this paper. The third author [PXR] was partially supported by the National Science Foundation grant number DMS-148695 through the Center for Undergraduate Research in Mathematics (CURM). The fifth author [IMS] was supported in part by Faculty Research funding from the Division of Science and Mathematics, New York University Abu Dhabi.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2019/11/15

Y1 - 2019/11/15

N2 - In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n×n matrix A, which counts the maximal number of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range W(A). We refer to k(A) as the Gau–Wu number of the matrix A. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify k(A). This continues the work of Wang and Wu [14] classifying the Gau–Wu numbers for 3×3 matrices. Our focus on singularities is inspired by Chien and Nakazato [3], who classified W(A) for 4×4 unitarily irreducible A with irreducible base curve according to singularities of that curve. When A is a unitarily irreducible n×n matrix, we give necessary conditions for k(A)=2, characterize k(A)=n, and apply these results to the case of unitarily irreducible 4×4 matrices. However, we show that knowledge of the singularities is not sufficient to determine k(A) by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different k(A). In addition, we extend Chien and Nakazato's classification to consider unitarily irreducible A with reducible base curve and show that we can find corresponding matrices with identical base curve but different k(A). Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky [5], generalizing their result in the process.

AB - In 2013, Gau and Wu introduced a unitary invariant, denoted by k(A), of an n×n matrix A, which counts the maximal number of orthonormal vectors xj such that the scalar products 〈Axj,xj〉 lie on the boundary of the numerical range W(A). We refer to k(A) as the Gau–Wu number of the matrix A. In this paper we take an algebraic geometric approach and consider the effect of the singularities of the base curve, whose dual is the boundary generating curve, to classify k(A). This continues the work of Wang and Wu [14] classifying the Gau–Wu numbers for 3×3 matrices. Our focus on singularities is inspired by Chien and Nakazato [3], who classified W(A) for 4×4 unitarily irreducible A with irreducible base curve according to singularities of that curve. When A is a unitarily irreducible n×n matrix, we give necessary conditions for k(A)=2, characterize k(A)=n, and apply these results to the case of unitarily irreducible 4×4 matrices. However, we show that knowledge of the singularities is not sufficient to determine k(A) by giving examples of unitarily irreducible matrices whose base curves have the same types of singularities but different k(A). In addition, we extend Chien and Nakazato's classification to consider unitarily irreducible A with reducible base curve and show that we can find corresponding matrices with identical base curve but different k(A). Finally, we use the recently-proved Lax Conjecture to give a new proof of a theorem of Helton and Spitkovsky [5], generalizing their result in the process.

KW - 4×4 matrices

KW - Boundary generating curve

KW - Field of values

KW - Gau–Wu number

KW - Irreducible

KW - Numerical range

KW - Singularity

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U2 - 10.1016/j.laa.2019.07.005

DO - 10.1016/j.laa.2019.07.005

M3 - Article

AN - SCOPUS:85068905445

VL - 581

SP - 112

EP - 127

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -