Home > ELA > Vol. 35 (2019)

#### Article Title

#### Keywords

Doubly substochastic matrices, Sub-defect, Maximum diagonal sum

#### Abstract

Let $\Omega_n$ denote the convex polytope of all $n\times n$ doubly stochastic matrices, and $\omega_{n}$ denote the convex polytope of all $n\times n$ doubly substochastic matrices. For a matrix $A\in\omega_n$, define the sub-defect of $A$ to be the smallest integer $k$ such that there exists an $(n+k)\times(n+k)$ doubly stochastic matrix containing $A$ as a submatrix. Let $\omega_{n,k}$ denote the subset of $\omega_n$ which contains all doubly substochastic matrices with sub-defect $k$. For $\pi$ a permutation of symmetric group of degree $n$, the sequence of elements $a_{1\pi(1)},a_{2\pi(2)}, \ldots, a_{n\pi(n)}$ is called the diagonal of $A$ corresponding to $\pi$. Let $h(A)$ and $l(A)$ denote the maximum and minimum diagonal sums of $A\in \omega_{n,k}$, respectively. In this paper, existing results of $h$ and $l$ functions are extended from $\Omega_n$ to $\omega_{n,k}.$ In addition, an analogue of Sylvesters law of the $h$ function on $\omega_{n,k}$ is proved.

#### Recommended Citation

Cao, Lei; Chen, Zhi; Duan, Xuefeng; Koyuncu, Selcuk; and Li, Huilan.
(2019),
"Diagonal Sums of Doubly Substochastic Matrices",
*Electronic Journal of Linear Algebra*,
Volume 35, pp. 42-52.

DOI: https://doi.org/10.13001/1081-3810.3760

*Abstract*