Linear maps on nonnegative symmetric matrices preserving a given independence number

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace, and let $J_n$ be the $n\times n$ matrix with all entries equal to one. Given any integers $n,t$ with $2\le t\le n-1$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $\phi(J_n-I_n)=J_n-I_n$ and $$\alpha(\phi(X))=t~\iff \alpha(X)=t {~~~~\rm for~ all~}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=P^TXP{~~~~\rm for~ all~}X\in S_n^+.$

Preserver Problems on Matrices

Preserver problems on matrices concern the characterization of linear or nonlinear maps or operators on matrices that preserve properties of the space of matrices or leave certain functions, subsets, and relations invariant. In this talk, I will present some results on both linear and nonlinear preserver problems on matrices

Linear rank preservers of tensor products of rank one matrices

Let $n_1,\ldots,n_k$ be integers larger than or equal to 2. We characterize linear maps $\phi: M_{n_1\cdots n_k}\rightarrow M_{n_1\cdots n_k}$ such that $${\mathrm rank}\,(\phi(A_1\otimes \cdots \otimes A_k))=1\quad\hbox{whenever}\quad{\mathrm rank}\, (A_1\otimes \cdots \otimes A_k)=1 \quad \hbox{for all}\quad A_i \in M_{n_i},\, i = 1,\dots,k.$$ Applying this result, we extend two recent results on linear maps that preserving the rank of special classes of matrices.Comment: 12 page

The stable index of digraphs

The stable index of a digraph $D$ is defined to be the smallest integer $k$ such that $D$ contains two distinct $(k+1)$-walks with the same initial vertex and terminal vertex if such an integer exists; otherwise the stable index of $D$ is defined to be $\infty$. We characterize the set of stable indices of digraphs with a given order

Linear preservers and quantum information science

Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$. Suppose $|\cdot|$ is the Ky Fan $k$-norm with $1 \le k \le mn$, or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$. It is shown that a linear map $\phi: M_{mn} \rightarrow M_{mn}$ satisfying $$|A\otimes B| = |\phi(A\otimes B)|$$ for all $A \in M_m$ and $B \in M_n$ if and only if there are unitary $U, V \in M_{mn}$ such that $\phi$ has the form $A\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))V$, where $\varphi_i(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes ... \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.Comment: 13 page