Sign Patterns Of Nonnegative Matrix Power Sequences And A Class Of Invertible Matrices

In this thesis, we study the following three problems:(1) characterize those sign patterns of square nonnegative matrices A such that the sequence{f(Ak)}k=1∞ is nondecreasing (or nonincreasing), where f denotes the number of positive entries of a matrix; (2) study those invertible matrices A=(αij) ∈ Mn satisfying A-1= (αij-1) and those invertible matrices A= (αij) ∈ Mn satisfying{AT)-1= (αij-1), (We assume that 0-1=0); (3) study those integer matrices A satisfying Ak=I where k is a certain integer and I is the identity matrix. We obtain partial results for the three problems in our thesis.For problem (1), the number of positive entries of a nonnegative matrix does not change if we change the value of positive entries. Therefore, the problem can be simplified to studying 0-1 matrices. From the aspect of the rank of a matrix, we characterize those 0-1 matrices A with rank 1 and symmetric 0-1 matrices of rank 2 such that{f(Ak)}k=1∞is a monotonic sequence; From the aspect of the number of positive entries of a matrix, we characterize those 0-1 matrices A with f(A)< 3 and f(A)≥n2-n-1;in addition, we give several kinds of matrices which meet or do not meet the condition. For problem (2). we characterize those invertible tridiagonal matrices A=(αij) ∈ Mn satisfying A-1= (αij-1) and all the possible matrices of order 4 which meet the condition; we also characterize those invertible matrices A=(αij) ∈ Mn satisfying(AT)-1=(αij-1). For problem (3), we give a necessary and sufficient condition for A2= I when A is a tridiagonal integer matrix and a circulant matrix of an odd order.