Range decompositions and generalized square roots of positive semidefinite matrices

AbstractWe establish new connections between the range of a positive semidefinite matrix and its expressions as a finite positive linear combination of Hermitian projections. In particular, if Q is a positive semidefinite matrix and P a Hermitian projection onto any subspace of the range of Q, we provide a method for explicitly calculating the maximal r for which Q − rP is positive semidefinite