Linear transformations on matrices: the invariance of generalized permutation matrices—III

AbstractLet F be a field, F∗ be its multiplicative group, and H = {H:H is a subgroup of F∗ and there do not exist a, bϵF∗ such that Ha+b⊆H}. Let Dn be the dihedral group of degree n, H be a nontrivial group in H, and τn(H) = {α = (α1, α2,…, αn):αiϵH}. For σϵDn and αϵτn(H), let P(σ, α) be the matrix whose (i,j) entry is αiδiσ(j) (i.e., a generalized permutation matrix), and P(Dn, H) = {P(σ, α):σϵDn, αϵτn(H)}. Let Mn(F) be the vector space of all n×n matrices over F and TP(Dn, H) = {T:T is a linear transformation on Mn (F) to itself and T(P(Dn, H)) = P(Dn, H)}. In this paper we classify all T in TP(Dn, H) and determine the structure of the group TP(Dn, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper