Export Mapping Of Holding Invariants For Hermite Matrix Spaces

Suppose D is a division ring.αâ†'α is a involutorial anti automorphism from D to itself.A n x n matrix A=(αij),if A= (αij)and if AT is the transpose matrix of A over D.If (A)T=A,then A is called a Hermite matrix over D.Let n≥ be integers.Denote by Hn(D) the set of all n x n Hermite matrices over D,and fij(i,j ∈ [n])(i,j ∈ [n]) be maps from D to itself,here[n]={1,2, …,n}. We define the map: f:Hn(D)â†'Hn(D), by f:(xij)â†'fij(xij),(?)(xij) E Hn(D). then we say f is induced by{fij},Abbreviated to as"f is induced mapping"For any A ∈Hn(D),If A is invertible then f(A) is invertible and A is not invertible implies f(A) is not invertible,we say f is preserving invertible in both directions.If rankA+rankB= rank(A+B) implies rankf (A)+rankf(B) rank(f(A)+f(B)),we say f is rank-additivity preservers.Using the form of induced maps preserving rank-1 on Hn(D),in this article we determine the form of induced maps preserving invertibility in both directions on Hn(D), the form of induced maps preserving rank-additivity on Sn(F)and the form of induced maps preserving adjoint on Hn(F).