三幂等符号模式矩阵的结构

Abstract. A matrix whose entries are , -, and 0 is called a sign pattern matrix. For a signpattern matrix A,if A3 =A, then A is said to be sign tripotent. In this paper, the characteriza-tion of the n by n(n≥2) sign pattern matrices A which are sign tripotent has been given out.Furthermore, the necessary and sufficient condition of A3=A but A2≠A is obtained, too.     

Primitive non-powerful sign pattern matrices with base 2

A sign pattern matrix M with zero trace is primitive non-powerful if for some positive integer k, M ^k ?=?J _#. The base l(M) of the primitive non-powerful matrix M is the smallest integer k. By considering the signed digraph S whose adjacent matrix is the primitive non-powerful matrix M, we will show that if l(M)?=?2, the minimum number of non-zero entries of M is 5n???8 or 5n???7 depending on whether n is even or odd.     

The base sets of quasi-primitive zero-symmetric sign pattern matrices with zero trace

Cheng and Liu [Bo Cheng, Bolian Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715-731] showed that the base set of quasi-primitive zero-symmetric (generalized) sign pattern matrices is {1,2,…,2n}. The matrices with zero trace play a prominent role in matrix theory. In this paper, we investigate the bases of quasi-primitive zero-symmetric (generalized) sign pattern matrices with zero trace and prove that the base set of such matrices is {2,3,…,2n-1}.