Solutions of the Matrix Equation $XAX = X$, and Relations between Oblique and Orthogonal Projectors
Thomas N. E. Greville · SIAM Journal on Applied Mathematics · 1974
For a given $m \times n$ complex matrix A, it is shown that X satisfies $XAX = X$ if and only if it is expressible in the form \[ X = (EAF)^\dag \], where E and F are Hermitian idempotents and the dagger denotes the Moore–Penrose inverse. In particular, a matrix is idempotent if and only if it is the Moore–Penrose inverse of the product of two Hermitian idempotents: (The “if” part of the latter statement was previously shown by Cline.) If L and M are complementary subspaces of the space of n-dimensional complex vectors, and if $P_{L.M} $ denotes the projector on L along M and $P_L $ the orthogonal projector on L, it is shown that \[ P_{L,M} = \left(P_{M^ \bot } P_L \right)^\dag = \left( \left( I - P_M \right) P_L \right)^\dag \], where $M^ \bot $ denotes the orthogonal complement of M. More generally, \[ P_{L,M} = P_L Y\left( I - P_M \right) \], where Y is an arbitrary matrix satisfying $AYA = A$ with $A = ( I - P_M )P_L $. Afriat has previously shown that \[ P_{L,M} = \left( I - P_L P_M \right) ^{ - 1} P_L \left( I - P_L P_M \right) \] Here it is shown further that \[ P_{L.M} = \left( I - P_M P_L \right) ^{ - 1} \left( I - P_M \right) = P_L \left( P_L + P_M - P_M P_L \right) ^{ - 1} \]