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)).The matrix,A BD 1C, is called theSchur ComplementofDinM. IfAis invertible,then by eliminatingxfirst using the first equation we find that the Schur Complement ofAinMisD CA 1B(this corresponds to the Schur Complement defined in Boyd andVandenberghe [1] whenC=B>).

The Schur Complement and Symmetric Positive Semide nite (and De nite) Matrices Jean Gallier August 24, 2019 1 Schur Complements In this note, we provide some details and proofs of some results from Appendix A.5 (especially Section A.5.5) of Convex Optimization by Boyd and Vandenberghe [1]. Let Mbe an n nmatrix written a as 2 2 block matrix M= A ...

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

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