Two Additions To The Symmetric Matrix Space

Nearly twenty years.there have been many studies about preserving additive maps on matrix including the invertibility preserving and rank equation preserving are two important aspects.In addition to the existing research,in this paper,we study the two preserving problems on the space of alternate matrices over the field F with chF≠2.One is the invertibility preserving.the other is Sylvester rank equation preserving.Suppose chF≠2,n≥4and n be even number.Kn(F) is called the space of n×n alternate matrices over F.φ is the map from Kn(F) to itself.If A∈Kn(F) is invertible if and only if φ(A)∈Kn(F) is invertible,then call φ preserves the two-way invertibility.For A,B∈Kn(F),r(A) is denoted by the rank of A.If r(AB)=r(A)+r(B)-n if and only if r(φ(A)φ(B))=r(φ(A))+r(φ(B))-n,then call φ preserves the two-way Sylvester rank equation.First, this paper gives the form of additive surjection from Kn(F) to itself,which preserves the two-way invertibility.On the basic of preserving invertibility,the paper gives the form of additive surjection from Kn(F) to itself,which preserves the two-way Sylvester rank equation.