A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D_{1} and D_{2} such that A^{-T} = D_{1}AD_{2}, where A^{-T} denotes the transpose of the inverse of A. Denote by J = diag(±1) a diagonal (signature) matrix, each of whose diagonal entries is +1 or −1. A nonsingular real matrix Q is called J-orthogonal if Q^{T} JQ = J. Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation of a J-orthogonal matrix. An investigation into the sign patterns of the J-orthogonal matrices is initiated. It is observed that the sign patterns of the G-matrices are exactly the column permutations of the sign patterns of the J-orthogonal matrices. Some interesting constructions of certain J-orthogonal matrices are exhibited. It is shown that every symmetric staircase sign pattern matrix allows a J-orthogonal matrix. Sign potentially J-orthogonal conditions are also considered. Some examples and open questions are provided., Frank J. Hall, Miroslav Rozložník., and Obsahuje seznam literatury
Let M_{m,n} be the set of all m × n real matrices. A matrix A \in M_{m,n} is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_{m,n} \rightarrow M_{m,n} that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A \in M_{m,n} is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found., Sara M. Motlaghian, Ali Armandnejad, Frank J. Hall., and Obsahuje seznam literatury