2023/09/16 by Mikić, Jovan
#15B35 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2309.08933
We consider a new class of matrices associated to a real square matrix A and to a vector c ∈ \-1,1\n such that c1=1 by using a map φ_c which turns out to be a conjugation of a matrix A by a signature matrix. It is shown that every such matrix is similar and congruent to a matrix A and that they have same permanental polynomials. There are 2n-1 maps φ_c and they form an abelian group under the composition of maps isomorphic to the group (ℤ2n-1, +). A decomposition of matrices, on a symmetric and antisymmetric matrix under a map φ_c, is considered. Particularly, it is shown that sum of all principal minors of the order two of a matrix A is equal to the sum of all principal minors of the order two of their symmetric and antisymmetric parts. It is shown that any symmetric matrix and any antisymmetric matrix under the map φ_c are simultaneously permutation similar to certain block matrices which have two blocks. Finally, for a fixed matrix A, it is proved that the number of different matrices φ_c(A) is 2n-t, where t is the number of connected components of the graph G whose adjacency matrix is A.