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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. Denote by $J_n$ the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $$\alpha(\phi(X))= \alpha(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=H\circ(P^TXP)\quad { \rm for~ all\quad}X\in S_n^+,$$ where $H=\phi(J_n-I_n)$ with all off-diagonal entries positive.

In this paper, we determine the maximum number of nonzero entries in 0-1 matrices of order $n$ with zero trace whose squares are 0-1 matrices when $n\ge 8$. The extremal matrices attaining this maximum number are also characterized., Comment: 20 pages, 11 figures

We prove that a nonsymmetric normal entry pattern of order n ( n ≥ 3 ) has at most n ( n − 3 ) / 2 + 3 distinct indeterminates and up to permutation similarity this number is attained by a unique pattern which is explicitly described. An open problem is posed.

Let F m × n be the set of m × n matrices over a field F . Consider a graph G = ( F m × n , ∼ ) with F m × n as the vertex set such that two vertices A , B ∈ F m × n are adjacent if rank ( A − B ) = 1 . We study graph properties of G when F is a finite field. In particular, G is a regular connected graph with diameter equal to min { m , n } ; it is always Hamiltonian. Furthermore, we determine the independence number, chromatic number and clique number of G. These results are used to characterize the graph endomorphisms of G, which extends Hua's fundamental theorem of geometry on F m × n .

We firstly consider the block dominant degree for I-(II-)block strictly diagonally dominant matrix and their Schur complements, showing that the block dominant degree for the Schur complement of an I-(II-)block strictly diagonally dominant matrix is greater than that of the original grand block matrix. Then, as application, we present some disc theorems and some bounds for the eigenvalues of the Schur complement by the elements of the original matrix. Further, by means of matrix partition and the Schur complement of block matrix, based on the derived disc theorems, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and the numerical example illustrates that the iteration can compute out the results faster.

We characterize the ACI-matrices all of whose completions have the same rank, determine the largest number of indeterminates in such partial matrices of a given size, and determine the partial matrices that attain this largest number.

In this paper, we obtain some estimates for the γ-diagonally and product γ-diagonally dominant degree of the Schur complement of matrices. As application we present some bounds for the eigenvalues of Schur complement by the entries of the original matrix. Based on these results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster.

We show that for every invertible n × n complex matrix A there is an n × n diagonal invertible D such that AD has distinct eigenvalues. Using this result, we affirm a conjecture of Feng, Li, and Huang that an n × n matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n - 1 are zero.

We characterize the square partial matrices over a field all of whose completions have the same spectrum, and determine the maximum number of indeterminates in such partial matrices of a given order as well as the matrices that attain this maximum number.

A sign pattern is said to be nilpotent of index k if all real matrices in its qualitative class are nilpotent and their maximum nilpotent index equals k. In this paper, we characterize sign patterns that are nilpotent of a given index k. The maximum number of nonzero entries in such sign patterns of a given order is determined as well as the sign patterns with this maximum number of nonzero entries.

We prove the spectral radius inequality ρ ( A 1 ∘ A 2 ∘ ⋯ ∘ A k ) ⩽ ρ ( A 1 A 2 ⋯ A k ) for nonnegative matrices using the ideas of Horn and Zhang. We obtain the inequality ‖ A ∘ B ‖ ⩽ ρ ( A T B ) for nonnegative matrices, which improves Schur’s classical inequality ‖ A ∘ B ‖ ⩽ ‖ A ‖ ‖ B ‖ , where ‖ · ‖ denotes the spectral norm. We also give counterexamples to two conjectures about the Hadamard product.