This post concerns the geometry of the convex cone of positive definite symmetric matrices. Let us denote by \\( {\\mathcal{S}_n} \\) the set of \\( {n\\times n} \\) real symmetric matrices, and by \\( {\\mathcal{S}_n^+} \\) the subset of positive definite matrices. Let us fix some \\( {K\\in\\mathcal{S}_n} \\) and write<\/p>\n
\\[ K:= \\begin{pmatrix} A & W \\\\ W^\\top & S \\end{pmatrix} \\]<\/p>\n
with \\( {A\\in\\mathcal{S}_{n-m}} \\) and \\( {S\\in\\mathcal{S}_{m}} \\) for some fixed \\( {1\\leq m<n} \\). If \\( {A\\in\\mathcal{S}_{n-m}^+} \\) then the Schur complement<\/b> of \\( {A} \\) with respect to \\( {K} \\) is the matrix<\/p>\n
\\[ T:=S-W^\\top A^{-1}W\\in\\mathcal{S}_m. \\]<\/p>\n
It is well known that<\/p>\n
\\[ K\\in\\mathcal{S}_n^+ \\quad\\text{iff}\\quad A\\in\\mathcal{S}_{n-m}^+ \\quad\\text{and}\\quad T\\in\\mathcal{S}_m^+, \\]<\/p>\n
and moreover, \\( {K^{-1}} \\) admits the block decomposition<\/p>\n
\\[ K^{-1} = \\begin{pmatrix} I_n & -A^{-1}W \\\\ 0 & I_m \\\\ \\end{pmatrix} \\begin{pmatrix} A^{-1} & 0 \\\\ 0 & T^{-1} \\\\ \\end{pmatrix} \\begin{pmatrix} I_n & 0 \\\\ -W^\\top A^{-1} & I_m \\\\ \\end{pmatrix}, \\]<\/p>\n
see for instance Theorem 7.7.6 page 472 in Horn and Johnson, Matrix analysis, Cambridge University Press, 1990<\/a>. In terms of Gaussian random vectors, if \\( {X=(Y,Z)} \\) is a Gaussian random vector of \\( {\\mathbb{R}^n=\\mathbb{R}^{n-m}\\times\\mathbb{R}^m} \\) with \\( {1\\leq m<n} \\), with covariance matrix \\( {K} \\), then<\/p>\n \\[ Y\\sim\\mathcal{N}(0,A) \\quad\\text{and}\\quad Z\\sim\\mathcal{N}(0,S) \\quad\\text{and}\\quad \\mathcal{L}(Z\\,\\vert\\,Y)=\\mathcal{N}(W^\\top A^{-1}Y,T). \\]<\/p>\n More generally, for a non empty \\( {I\\subset\\{1,\\ldots,n\\}} \\), we have, with \\( {J:=I^c} \\),<\/p>\n \\[ X_I\\sim\\mathcal{N}(0,K_{I,I}) \\quad\\text{and}\\quad X_{J}\\sim\\mathcal{N}(0,K_{J,J}) \\]<\/p>\n and the Schur complement of \\( {K_{J,J}} \\) in \\( {K} \\) is \\( {K_{I,I}-K_{I,J}(K_{J,J})^{-1}K_{J,I}} \\) and<\/p>\n \\[ \\mathcal{L}(X_I\\,\\vert\\,X_{J}) \\sim\\mathcal{N}(K_{I,J}(K_{J,J})^{-1}X_{J},\\,K_{I,I}-K_{I,J}(K_{J,J})^{-1}K_{J,I}). \\]<\/p>\n Moreover, from the block inverse formula we get<\/p>\n \\[ (K^{-1})_{I,I} = (K_{I,I}-K_{I,J}(K_{J,J})^{-1}K_{J,I})^{-1}. \\]<\/p>\n The components of \\( {X_I} \\) are conditionally independent<\/b> given the remaining components iff \\( {((K^{-1})_{I,I})^{-1}} \\) is diagonal, i.e. iff \\( {(K^{-1})_{I,I}} \\) is diagonal. In particular, if \\( {I=\\{i,j\\}} \\) with \\( {i\\neq j} \\) then \\( {X_i} \\) and \\( {X_j} \\) are conditionally independent given the remaining components iif \\( {(K^{-1})_{i,j}=0} \\).<\/p>\n The extremal elements of the closed convex cone \\( {\\overline{\\mathcal{S}_n^+}} \\) are the unit rank matrices \\( {\\{uu^\\top: u\\in\\mathbb{R}^n\\}} \\). Every \\( {A\\in\\overline{\\mathcal{S}_n^+}} \\) can be written as \\( {A=\\lambda_1 u_1u_1^\\top+\\cdots+\\lambda_n u_nu_n^\\top} \\) where \\( {\\lambda_1,\\ldots,\\lambda_n} \\) and \\( {u_1,\\ldots,u_n} \\) denote the spectrum and the eigenvectors of \\( {A} \\) respectively. This is the Carathéodory local parametrization of the convex cone \\( {\\overline{\\mathcal{S}_n^+}} \\) with \\( {n} \\) Carathéodory conic rays \\( {\\lambda_1 u_1u_1^\\top,\\ldots,\\lambda_r u_nu_n^\\top} \\).<\/p>\n For any \\( {K\\in\\mathcal{S}_n} \\), any \\( {1\\leq i\\leq j\\leq n} \\), and any \\( {x\\in\\mathbb{R}} \\), let \\( {\\xi_{i,j}(K,x)} \\) be the element of \\( {\\mathcal{S}_n} \\) obtained from \\( {K} \\) by replacing \\( {K_{i,j}} \\) and \\( {K_{j,i}} \\) by \\( {x} \\). In particular,<\/p>\n \\[ K=\\xi_{i,j}(K,K_{i,j}). \\]<\/p>\n Now for which values of \\( {x} \\) the matrix \\( {\\xi_{i,j}(K,x)} \\) belongs to \\( {\\mathcal{S}_n^+} \\)? The answers is as follows.<\/p>\n