The Schur Complement and Symmetric Positive Semide …

The Schur Complement and Symmetric PositiveSemidefinite (and Definite) MatricesJean GallierAugust 24, 20191 Schur ComplementsIn this note, we provide some details and proofs of some results from Appendix (especiallySection ) ofConvex Optimizationby Boyd and Vandenberghe [1].LetMbe ann nmatrix written a as 2 2 block matrixM=(A BC D),whereAis ap pmatrix andDis aq qmatrix, withn=p+q(so,Bis ap qmatrixandCis aq pmatrix). We can try to solve the linear system(A BC D)(xy)=(cd),that isAx+By=cCx+Dy=d,by mimicking Gaussian elimination, that is, assuming thatDis invertible, we first solve forygettingy=D 1(d Cx)and after substituting this expression foryin the first equation, we getAx+B(D 1(d Cx)) =c,that is,(A BD 1C)x=c BD the matrixA BD 1 Cis invertible, then we obtain the solution to our systemx= (A BD 1C) 1(c BD 1)

so both MMyand MyM are orthogonal projections (since they are both symmetric). We claim that MMyis the orthogonal projection onto the range of Mand MyMis the orthogonal projection onto Ker(M)?, the orthogonal complement of Ker(M). Obviously, range(MMy) range(M) and for any y= Mx2range(M), as MMyM= M, we have MMyy= MMyMx= Mx= y;

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  Positive, Projection, Complement, Orthogonal, Crush, Demise, Symmetric, Orthogonal projection, Schur complement and symmetric positive semide

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