Sub-defect of product of doubly substochastic matrices

In this paper, we provide three different ways to partition the polytope of doubly substochastic matrices into subpolytopes via the prescribed row and column sums, the sum of all elements and the sub-defect respectively. Then we characterize the extreme points of each type of convex subpolytopes. The relations of the extreme points of the subpolytopes in the three partitions are also given.Denote the set of all n × n doubly stochastic matrices by Ω n , and the set of all n × n doubly substochastic matrices by ω n . The set Ω n is a convex polytope, and has been intensively studied by many mathematicians [11,12,13,15,16,17,18,23,24,25,33,34,32]. Specially, the extreme points of Ω n are exactly the permutation matrices due to Birkhoff [4] and von Neumann [31], which can be stated as follows.Theorem 1.3. [4, 31] An n × n matrix A is a doubly stochastic matrix if and only if there are finite permutation matrices P 1 , P 2 , · · · , P N and positive numbers α 1 , · · · , α N such that α 1 + · · · + α N = 1 and A = α 1 P 1 + · · · + α N P N .The set ω n is also a convex polytope and its extreme points are partial permutation matrices [21], i.e., matrices with at most one element in each row and each column equal to one and other elements zero. Since Ω n ⊆ ω n , one may wonder if the classical 2010 Mathematics Subject Classification. 15B51; 52B05; 05A18.

show abstract

“…The first way to partition ω n is induced by a characteristic of doubly substochastic matrices called sub-defect [7,6,5]. Definition 2.1.…”

Section: Three Different Partitions Of ω Nmentioning

confidence: 99%

Partitions of the polytope of doubly substochastic matrices

Cao

Zhi

2019

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is the smallest integer k such that there exists an (n + k) × (n + k) doubly stochastic matrix containing B as a submatrix. It has been shown that the sub-defect can be calculated easily by taking the ceiling of the difference of the size of the matrix and the sum of all entries (see [4], [5] and [6]). Diagonal Sums of Doubly Substochastic Matrices Theorem 1.1.…”

mentioning

confidence: 99%

“…Actually, if A, B ∈ ω n , then AB ∈ ω n (Proposition 2.4 in [5]). We can evaluate the extreme values of h(AB) and l(AB).…”

mentioning

confidence: 99%

Diagonal Sums of Doubly Substochastic Matrices

Cao

Chen

Duan

et al. 2019

ELA

Self Cite

View full text Add to dashboard Cite

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.

show abstract