## Mathematics Faculty Articles

#### Document Type

Article

#### Publication Date

2-2019

#### Publication Title

Electronic Journal of Linear Algebra

#### Keywords

Doubly substochastic matrices, Sub-defect, Maximum diagonal sum

#### ISSN

1081-3810

#### Volume

35

#### First Page

45

#### Last Page

52

#### Abstract

Let Ω* _{n}* denote the convex polytope of all

*n x n*doubly stochastic matrices, and ω

*denote the convex polytope of all*

_{n}*n x n*doubly substochastic matrices. For a matrix

*A*ϵ ω

*, define the sub-defect of*

_{n}*A*to be the smallest integer

*k*such that there exists an (

*n + k*) x (

*n + k*) doubly stochastic matrix containing

*A*as a submatrix. Let ω

*denote the subset of ω*

_{n,k}*which contains all doubly substochastic matrices with sub-defect*

_{n}*k.*For

*π*a permutation of symmetric group of degree

*n*, the sequence of elements

*a*

_{1π(1)};

*a*

_{2π(2)}, ...,

*a*

_{nπ(n)}is called the diagonal of

*A*corresponding to

*π*. Let

*h*(

*A*) and

*l*(

*A*) denote the maximum and minimum diagonal sums of

*A*ϵ ω

*, respectively. In this paper, existing results of*

_{n,k}*h*and

*l*functions are extended from Ω

*to ω*

_{n}*In addition, an analogue of Sylvesters law of the*

_{n,k.}*h*function on ω

*is proved.*

_{n,k}#### Additional Comments

National Natural Science Foundation of China grant #s: 11601233, 11561015; Fundamental Research Funds for the Central Universities grant #: KJQN201718; Natural Science Foundation of Jiangsu Province grant #: BK20160708; Natural Science Foundation of Guangxi Province grant #: 2016GXNSFFA380009

#### NSUWorks Citation

Cao, Lei; Chen, Zhi; Duan, Xuefeng; Koyuncu, Selcuk; and Li, Huilan, "Diagonal Sums of Doubly Substochastic Matrices" (2019). *Mathematics Faculty Articles*. 279.

https://nsuworks.nova.edu/math_facarticles/279

#### ORCID ID

0000-0001-7613-7191

#### ResearcherID

G-7341-2019

#### DOI

10.13001/1081-3810.3760

## Comments

This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository.