Sparsity Of Matrices And Zero-Nonzero Patterns

We study some problems on sparsity of matrices, zero-nonzero patterns, sign patterns and matrix powers. Our main results are as follows.1. Let F be a field and a1,a2,..., an be distinct indeterminates. Let A be a matrix of order n whose entries are rational functions of a1,a2,..., an over F. Assume that the characteristic polynomial of A isp(x)=xn+aixnn-1+…+an-1x+an. By using transcendence degrees of field extensions in algebra and properties of spanning branchings in graph theory, we prove that the number of nonzero entries in A is larger than or equal to2n-1. Denote by C(p) the companion matrix of p(x). Since the entries of C(p) are rational functions of a1, a2,……,an over F and C(p) has exactly2n-1nonzero entries, the companion matrix is the sparsest in this sense.2. Let F be a field with at least three elements. We determine the irreducible zero-nonzero patterns A such that for any nonsingular matrix B over F with pattern A, the inverse B-1has the same pattern A. We also determine the zero-nonzero patterns P such that for any nonsingular matrix Q over F with pattern P, the transpose of the inverse (Q-1)T has the same pattern P.3. Let F be a field with at least three elements. We characterize the zero-nonzero patterns A such that for any matrix B over F with pattern A, B2has the same pattern A. Then we determine the possible numbers of nonzero entries in such patterns with a given minimum rank as well as the patterns that attain the maximum number. The results can be stated in terms of idempotent0-1matrices. 4. We characterize idempotent Toeplitz sign patterns and idempotent Han-kel sign patterns. This work extends some existing results.5. We characterize the matrices whose powers are eventually diagonal, Toeplitz and normal respectively.