ON SUB-DEFECT OF HADAMARD PRODUCT OF DOUBLY SUBSTOCHASTIC MATRICES

Abstract

The \emph{sub-defect} of $A$, defined as $\mathrm{sd}(A) = \lceil n - \mbox{sum}(A) \rceil,$ is the minimum number of rows and columns required to be added to transform a doubly substochastic matrix into a doubly stochastic matrix. Here, $n$ is the size of $A$ and $\mbox{sum}(A)$ is the sum of all entries of matrix $A.$ In this paper, we show that for arbitrary doubly substochastic matrices $A$ and $B$, the Hadamard product $A \circ B$ is also a doubly substochastic matrix, and $ \max \{sd(A),sd(B)\}\leq sd(A \circ B) \leq \max\{n, sd(A)+sd(B)\}. $