Staggered Array Of Linear Preserver Problems

Let F be a field, R be a real field, this paper mainly studys two linear preserver problems on alternate matrices. When charF 2, alternate matrices is skew-symmetric matrices. Let SKn(F) be the vector space of n x n skew-symmetric matrices over the field F, Mn(F) be the vector space of n x n matrices over the field .F, Tn(F) be the vector space of n x n upper triangular matrices over the field F.On one hand,the mapping preserving determinant is characterised from Mn(F) to Mn(F) and from Tn(F) to Tn(F) in reference [43], however,in this paper, we characterise the forms of mapping preserving determinant from SKn(R) to SKn(K); on the other hand.when F is an infinite field with charF ^ 2 and n is a positive even,the linear mapping preserving adjugate from SKn(F) to SKn(F) is characterised in reference [7],however,in this paper,when char F ^ 2, n is a positive even,we characterise the forms of linear mapping preserving adjugate from SKn(F) to SKn(F).