The Further Research Of K-idempotent Matrices

In this thesis, we first study the relation of the range and the null space between twok-idempotent matrix T₁,T₂ and the linear combination of the form T = c₁T₁+c₂T₂. Alsothe problem of when T is nonsingular and group invertible are considered. We deriveexplicit formulace of the inverse and the group inverse of T under some conditions.We derive a very short expression for the group inverse of a₁ +···+ a_n, wherea₁,···,a_n are elements in Banach algebra having group inverse and satisfying a_ia_j = 0for i < j. Then we apply this formula in order to find the group inverse of 2×2 blockoperators under some conditions.We obtained the su?cient and necessary conditions that the linear combinationof hyper-generalized k-idempotent matrices H = c₁H₁ + c₂H₂ is also hyper-generalizedk-idempotent matrix.