Is This a New Class of Matrices?

We consider a new class of matrices associated with a real square matrix [Formula: see text] and a vector [Formula: see text] such that [Formula: see text] by using a map [Formula: see text] which turns out to be a conjugation of a matrix [Formula: see text] by a signature matrix. It is shown that every such matrix is similar and congruent to a matrix [Formula: see text] and that they have the same permanental polynomials. There are [Formula: see text] maps [Formula: see text] and they form an abelian group under the composition of maps isomorphic to the group [Formula: see text]. A decomposition of matrices into a sum of symmetric and antisymmetric part under a map [Formula: see text] is considered. Particularly, it is shown that the sum of all principal minors of order [Formula: see text] of a matrix [Formula: see text] is equal to the sum of all principal minors of order [Formula: see text] of their symmetric and antisymmetric parts. It is shown that any symmetric matrix and any antisymmetric matrix under the map [Formula: see text] are simultaneously permutation similar to certain block matrices whose have [Formula: see text] blocks. Finally, for a fixed matrix [Formula: see text], it is proved that the number of different matrices [Formula: see text] is [Formula: see text], where [Formula: see text] is the number of connected components of the graph [Formula: see text] whose adjacency matrix is [Formula: see text].