### Abstract

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 x_{j} such that the scalar products 〈Ax_{j},x_{j}〉 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.

Original language | English (US) |
---|---|

Pages (from-to) | 112-127 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 581 |

DOIs | |

State | Published - Nov 15 2019 |

### Fingerprint

### Keywords

- 4×4 matrices
- Boundary generating curve
- Field of values
- Gau–Wu number
- Irreducible
- Numerical range
- Singularity

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*581*, 112-127. https://doi.org/10.1016/j.laa.2019.07.005

**Singularities of base polynomials and Gau–Wu numbers.** / Camenga, Kristin A.; Deaett, Louis; Rault, Patrick X.; Sendova, Tsvetanka; Spitkovsky, Ilya; Yates, Rebekah B.Johnson.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 581, pp. 112-127. https://doi.org/10.1016/j.laa.2019.07.005

}

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

AU - Yates, Rebekah B.Johnson

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

UR - http://www.scopus.com/inward/record.url?scp=85068905445&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068905445&partnerID=8YFLogxK

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 -