Range decompositions and generalized square roots of positive semidefinite matrices
Let A be a positive semidefinite matrix. The author presents a method for finding all expressions of A as a finite positive linear combination of self-adjoint projections (not necessarily mutually orthogonal), and characterizes those for which the ranges of the projections are orthogonal or linearly independent subspaces. This is achieved by establishing a correspondence between such expressions and the (possibly rectangular) matrices B satisfying \(BB^*=A\). In particular, any linearly independent decomposition of a projection is necessarily an orthogonal decomposition. If P is a rank one projection whose range is spanned by a unit vector v, then the maximal number r for which A-rP\(\geq 0\) is shown to be \((v,x)^{-1}\), where x is any solution to \(Ax=v\).
scientific article; zbMATH DE number 4189932
scientific article