An Explicit Form of the Moore–Penrose Inverse of an Arbitrary Complex Matrix

(2) AAtPR(A) = v (3) AtPR(A) At = 0 is the Moore-Penrose inverse of A [12]. An equivalent definition is given in [10]. As is well known, every A has one and only one At. 2. Results. For a given matrix B we denote by B{ 1 } the set B{ 1 } {X:BXB = B} . THEOREM. For any matrix A, (4) At = A*TA*, where Te A*AA*{1}. Proof. At = AtAAt = At(AAt)*A(AtA)*At = At(At)*A*AA*(At)*At = At(At)*A*AA*TA*AA*(At)*At = At(AAt)*AA*TA*A(AtA)*At = (AtA)A*TA*(AAt) = A*TA*. All these equalities follow directly from (1), (2) and (3). COROLLARY 1. Let A A(l A12) A2 1 A22 where A11 is a square nonsingular matrix such that rank Al = rank A. Then