### Abstract

For a given n× n matrix A, let k(A) stand for 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). This number was recently introduced by Gau and Wu and we therefore call it the Gau-Wu number of the matrix A. We compute k(A) for two classes of n × n matrices A. A simple and explicit expression for k(A) for tridiagonal Toeplitz matrices A is derived. Furthermore, we prove that k(A)=2 for every pure almost normal matrix A. Note that for every matrix A we have k(A)≥2, and for normal matrices A we have k(A)=n, so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the Gau-Wu number. Finally, matrices with maximal Gau-Wu number (k(A)=n) are considered.

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

Pages (from-to) | 254-262 |

Number of pages | 9 |

Journal | Linear Algebra and Its Applications |

Volume | 444 |

DOIs | |

State | Published - Mar 1 2014 |

### Fingerprint

### Keywords

- Almost normal matrices
- Numerical range
- Toeplitz matrices

### ASJC Scopus subject areas

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

### Cite this

*Linear Algebra and Its Applications*,

*444*, 254-262. https://doi.org/10.1016/j.laa.2013.11.045

**On the Gau-Wu number for some classes of matrices.** / Camenga, Kristin A.; Rault, Patrick X.; Sendova, Tsvetanka; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 444, pp. 254-262. https://doi.org/10.1016/j.laa.2013.11.045

}

TY - JOUR

T1 - On the Gau-Wu number for some classes of matrices

AU - Camenga, Kristin A.

AU - Rault, Patrick X.

AU - Sendova, Tsvetanka

AU - Spitkovsky, Ilya

PY - 2014/3/1

Y1 - 2014/3/1

N2 - For a given n× n matrix A, let k(A) stand for the maximal number of orthonormal vectors xj such that the scalar products «Ax j,xj» lie on the boundary of the numerical range W(A). This number was recently introduced by Gau and Wu and we therefore call it the Gau-Wu number of the matrix A. We compute k(A) for two classes of n × n matrices A. A simple and explicit expression for k(A) for tridiagonal Toeplitz matrices A is derived. Furthermore, we prove that k(A)=2 for every pure almost normal matrix A. Note that for every matrix A we have k(A)≥2, and for normal matrices A we have k(A)=n, so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the Gau-Wu number. Finally, matrices with maximal Gau-Wu number (k(A)=n) are considered.

AB - For a given n× n matrix A, let k(A) stand for the maximal number of orthonormal vectors xj such that the scalar products «Ax j,xj» lie on the boundary of the numerical range W(A). This number was recently introduced by Gau and Wu and we therefore call it the Gau-Wu number of the matrix A. We compute k(A) for two classes of n × n matrices A. A simple and explicit expression for k(A) for tridiagonal Toeplitz matrices A is derived. Furthermore, we prove that k(A)=2 for every pure almost normal matrix A. Note that for every matrix A we have k(A)≥2, and for normal matrices A we have k(A)=n, so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the Gau-Wu number. Finally, matrices with maximal Gau-Wu number (k(A)=n) are considered.

KW - Almost normal matrices

KW - Numerical range

KW - Toeplitz matrices

UR - http://www.scopus.com/inward/record.url?scp=84891630597&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84891630597&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2013.11.045

DO - 10.1016/j.laa.2013.11.045

M3 - Article

AN - SCOPUS:84891630597

VL - 444

SP - 254

EP - 262

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -