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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 1d)y=D 1(d C(A BD 1C) 1(c BD 1d)).
which has no solution unless Pand bsatisfy certain conditions. 3 Pseudo-Inverses We will need pseudo-inverses so let’s review this notion quickly as well as the notion of SVD which provides a convenient way to compute pseudo-inverses. We only consider the case of square matrices since this is all we need. For comprehensive treatments of SVD and
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