Asian Research Journal of Mathematics

The problems of characterizing maps that preserve certain invariant on given sets are called the preserving problems, which have become one of the core research areas in matrix theory. If for any A₁ ⊕···⊕ A_k’B₁ ⊕···⊕ B_k ∈ M _n1 ⊕···⊕ M_nk, a linear map, Φ : M_n1 ⊕···⊕ M_nk → M_n1 ⊕···⊕ M_nk , as R ₍A₁ ⊕···⊕ A_k + B₁ ⊕···⊕ B_{k) =} R ₍A₁ ⊕···⊕ A_k) + R (B₁ ⊕···⊕ B_k) established, there is R (Φ (A₁ ⊕···⊕ A_k + B₁ ⊕···⊕ B_K)) = R (Φ (A₁ ⊕···⊕ A_k)) + R (Φ(B₁ ⊕···⊕ B_K)) we say that  Φ preserves the rank-additivity. If for any A₁ ⊕···⊕ A_k′B₁ ⊕···⊕ B_k ∈ M_n1 ⊕···⊕ M_nk, and a linear map, Φ : M_n1 ⊕···⊕ M_nk → M_n1 ⊕···⊕ M_nk  , as established, there is  R(A₁ ⊕···⊕ A_k + B₁ ⊕···⊕ B_k) = |R(A₁( ⊕···⊕ A_k) — R (B₁ ⊕···⊕ B_k) we say that Φ rank-sum-miminal. In this paper, we characterize the form of linear mapping Φ.