SUBMATRIX OF ORTHOGONAL PROJECTION
Junlian Xu · Journal of South China Normal University · 2011
LetHbe an-dimensional complex Hilbert space,andQbe an orthogonal projection onH.IfMis asubspace ofHand dimM=r,then under the space decompositionH=M M⊥,one hasQ=A BB*D,whereAB(M),BB(M⊥,M),DB(M⊥).By using the technique of block operator matrix,the properties and relations be-tweenA,BandDare given,and the relations betweenPandQare discussed,wherePis an orthogonal projectiononM.Furthermore,the following conclusions are obtained:(i)R(P)∩R(Q) ={0} dimR(A) =dimR(B);(ii)R(P)+R(Q) =H dimR(D) =n-r;(iii)R(P)⊥R(Q) dimR(A) =0.