Linear preservers on triangular matrices
Let \(T_n(F)\) be the vector space of all \(n\times n\) upper triangular matrices over a field \(F\). Rank one preservers \(T\) on \(T_n(F)\) are characterized if they satisfy one of the following properties: (1) \(T\) preserves rank one matrices, (2) \(T\) preserves both singular and nonsingular matrices, (3) \(T\) is nonsingular and \(T(\text{adj } A)=\text{adj } T(A)\) for all \(A\in T_n(F)\). The determinant preserving linear transformations on \(T_n(F)\) is also characterized. Nonsingular linear transformations which commute with the adjugate function by using the structure of rank one preservers on \(T_n(F)\) are discussed.
scientific article; zbMATH DE number 1100773
scientific article