Range decompositions and generalized square roots of positive semidefinite matrices
type: Item
instance of: scholarly article
title: Range decompositions and generalized square roots of positive semidefinite matrices
zbMATH Open document ID: 0722.15023
DOI: 10.1016/0024-3795(91)90296-9
published in: Linear Algebra and its Applications
publication date: 1991-01-01T00:00:00Z
review text: 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 \(Ax=v\).
reviewed by: Jaroslav Zemánek
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 4189932
zbMATH Keywords: range decompositions
zbMATH Keywords: generalized square roots
zbMATH Keywords: positive semidefinite matrix
zbMATH Keywords: self-adjoint projections
zbMATH Keywords: orthogonal decomposition
author: Donald A. Pierce
MaRDI profile type: Publication
full work available at URL: 0024-3795(91)90296-9
OpenAlex ID: W2054604651
cites work: Convex combinations of projections
cites work: On sums of projections
cites work: Any Hermitian matrix is a linear combination of four projections
cites work: Range Decomposition and Generalized Inverse of Nonnegative Hermitian Matrices
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description: scientific article; zbMATH DE number 4189932