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.