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The inverse of a block matrix can be expressed through the inverse of one diagonal block and the inverse of its Schur complement. This reduces many deletion and conditioning identities to smaller matrix inversions.

For a nondegenerate isometry direction $y$ and initially zero two-form, the local T-dual fields in alpha-prime-one units are $g_{yy}^{\prime }=1/g_{yy}$, $g_{yi}^{\prime }=0$, $g_{ij}^{\prime }=g_{ij}-g_{yi}{g}_{yj}/g_{yy}$, $B_{yi}^{\prime }=g_{yi}/g_{yy}$ and $B_{ij}^{\prime }=0$. Gauging the isometry and eliminating its auxiliary connection derives these rules. The spectator metric is a Schur complement. The sign of the mixed two-form follows the multiplier and dual-coordinate orientation convention.

For a finite simple graph with at least one vertex, the complement theta number has the equivalent semidefinite program formulationsandThe Schur complement and $U=tZ-J$ prove the equivalence; any feasible $t$ is at least one, so division by $t$ is valid. For a graph with an edge, $U/(t-1)$ is the Gram matrix of a strict vector coloring. Consequently $\bar{\vartheta }(G)$ is the strict vector chromatic number, and an ordinary $k$-graph colouring gives $\bar{\vartheta }(G)\le k$ by assigning the colors the vertices of a regular simplex.

For gauge fixing with $F^a=\partial \cdot A^a+(n\cdot \partial {\varphi }^a)/\sqrt{2}$, integrating out the Nakanishi-Lautrup field produces gauge-scalar mixing. The Schur complement of the free gauge-field kernel cancels the extra $(n\cdot p{)}^2/2$ in the scalar kernel, leaving $⟨{\varphi }^a(p){\varphi }^b(-p)⟩={\delta }^{ab}/p^2$. This is independent of the gauge vector $n$, but still depends on the full momentum $p$.

In the intermediate-index range, if an attaching sphere $A_1$ meets a belt sphere $B_1$ at exactly one transverse point, handle slides can remove the intersections of every other attaching sphere with $B_1$. The dual slides then remove the intersections of $A_1$ with the other belt spheres. Each move uses a band to the unique pivot sheet and a local isotopy removing the old and new intersections. If $m_{ji}=A_j\cdot B_i$ and $\delta =m_{11}=\pm 1$, the remaining incidence block is $m_{ji}-m_{j1}{m}_{1i}/\delta$, the Schur complement of the pivot. Extreme-index cases need separate hypotheses.

An LDL decomposition of a real symmetric matrix iswhere $L$ is unit lower triangular and $D$ is diagonal. Symmetric elimination computes it one Schur complement at a time. The matrix is positive definite exactly when every diagonal pivot in $D$ is positive.

Work with $z$ in the upper half-plane. Write the Stieltjes matrix resolvents and their normalized matrix traces asThe minor trace is normalized by $N$, not by $N-1$. This sign convention is the negative of the convention $(zI-X_N{)}^{-1}$ used in the general resolvent of an operator article. Here $g$ is the Stieltjes transform of a measure of the empirical spectral measure, with kernel $(x-z{)}^{-1}$.

The diagonal entries of $X_N$ are zero, so the preceding Schur complement formula gives $G_{ii}=-1/(z+q_i)$. Taking the matrix trace, subtracting the comparison value $-1/(z+g)$, and combining fractions gives the resolvent self-consistency defectThe positive numerator sign is fixed by this subtraction. All denominators are nonzero in the upper half-plane, as the imaginary-part estimate in the next part shows. The identity is deterministic and does not use entry independence or moment assumptions.

Use block matrix elimination with the invertible square block $A$:The left multiplier is block triangular with identity diagonal blocks, so its determinant is $1$. Taking the determinant of the block-triangular product provesThe matrix $D-C{A}^{-1}B$ is the Schur complement of $A$. The proof needs $A$ invertible, but does not assume the Schur complement or the full matrix is invertible.

Use the standard Euclidean convention $e^{-S_E}$ and restore the curvature normalization explicitly. Let $\phi$ be the conventional dimensionless dilaton, whose worldsheet term isFor constant ${\phi }_0$, the Gauss-Bonnet theorem gives $S_{\phi }={\phi }_0\chi (\Sigma )$. A closed surface of genus $g$ has Euler characteristic $\chi =2-2g$, so its contribution is weighted byThusThis is the dilaton normalization and Euler-characteristic weighting of string perturbation theory. If the unit curvature coefficient in the printed expression is retained literally in this Euclidean convention, then $\phi =4\pi \Phi$ and $g_s=e^{4\pi {\Phi }_0}$. If $\Phi$ denotes the conventional dilaton, the usual omitted $1/(4\pi )$ is understood and $g_s=e^{{\Phi }_0}$. One must specify that normalization before exponentiating the field. Lorentzian overall signs are fixed consistently by Wick rotation.

For the sigma-model beta function, usewith vanishing two-form and dilaton. Perform the background field expansion of a string sigma model covariantly by setting $X={\exp }_{\bar{X}}\eta$. In Riemann normal coordinates, the quadratic fluctuation action isThe covariant derivative contains the pulled-back target connection. This expansion accounts for the connection vertices as well as the normal-coordinate metric expansion; keeping only a single metric tadpole would not give a covariant answer.

Integrating the Gaussian fluctuations gives $\frac{1}{2}\mathrm{Tr}\log \Delta$, whereContracting the curvature vertex with the short-distance propagator traces $R^a{}_{cad}$ to the Ricci tensor $R_{cd}$. In dimensional regularization at $d=2-ϵ$, the logarithmic integral is $\int d^{d}p/(2\pi {)}^d(p^2+{\mu }^2{)}^{-1}=1/(2\pi ϵ)+O(1)$. With the stated curvature convention the divergent effective action isIt is cancelled by the metric counterterm $\delta g_{ab}={\alpha }^{\prime }{R}_{ab}/ϵ$. Equivalently, write $g^{\mathrm{bare}}={\mu }^{-ϵ}(g+{\alpha }^{\prime }R/ϵ+\cdots )$. Differentiating the bare coupling at fixed value gives the one-loop metric beta function of a string sigma modelThis loop is a loop of two-dimensional fluctuation fields: its expansion is in $\alpha \prime$, not a change of worldsheet genus or an extra power of $g_s$. The sign here defines the renormalization scale to increase toward the ultraviolet. For a round target sphere, positive Ricci curvature makes its squared radius increase with that scale, consistent with the usual asymptotic freedom of the inverse-radius sigma-model coupling. With the printed kinetic coefficient $1/2$ taken literally, $2\pi {\alpha }^{\prime }=1$ and the leading coefficient is $R_{ab}/(2\pi )$. Requiring the metric beta function to vanish gives $R_{ab}=0$ at this order. Full bosonic-string Weyl consistency also requires the other anomaly coefficients, including the critical-dimension condition when the dilaton vanishes.

For T-duality, choose coordinates $(y,x^i)$ adapted to the isometry and use ${\alpha }^{\prime }=1$ units for the local Buscher rules. Assume $g_{yy}\ne 0$, with a spacelike isometry for the usual unitary circle duality. In light-cone worldsheet coordinates take the convention that $E=g+B$ multiplies ${\partial }_{+}{X}^a{\partial }_{-}{X}^b$. Gauge translations of $y$, replace ${\partial }_{\pm }y$ by $A_{\pm }$, gauge fix $y=0$, and impose flatness with a multiplier $\tilde{y}$. The relevant first-order Lagrangian isIntegrating out $\tilde{y}$ gives a locally pure-gauge connection and hence the original model. Instead integrate the last term by parts and solve the algebraic equations for $A_{\pm }$:Substitution givesReading its symmetric and antisymmetric parts yields the Buscher dual of a metric with vanishing two-form:The spectator metric is the Schur complement of $g_{yy}$. Reversing the orientation of $\tilde{y}$ reverses the displayed mixed two-form signs; our multiplier and $E=g+B$ conventions fix them.

The metric rules arise already from the classical algebraic elimination. For quantum equivalence the regulated Gaussian measure also produces the Buscher dilaton shiftFor a compact isometry the multiplier periodicity and flat-connection sectors implement the exchange of momentum and winding. On a circle, restoring dimensions gives $R^{\prime }={\alpha }^{\prime }/R$. Thus the calculation derives the local dual background, while the measure and global sectors explain how it becomes a string-theory duality.

Put $\Omega ={\Sigma }^{-1}$, the precision matrix. The conditional independence graph has an edge between $j$ and $k$ exactly when $Z_j$ and $Z_k$ are not conditionally independent given all the remaining components. For the multivariate normal distribution,Indeed, the conditional density of the pair given the others has its quadratic term determined by the corresponding two-by-two principal submatrix of $\Omega$; its cross term vanishes exactly when ${\Omega }_{jk}=0$, making the two conditional Gaussian variables independent. This is a Gaussian graphical model.

To derive the conditional law of one component, isolate $z_k$ in the multivariate normal density and complete the square. The terms involving it areThereforeEquivalently, its mean is ${\mu }_k+{\Sigma }_{k,-k}{\Sigma }_{-k,-k}^{-1}(z_{-k}-{\mu }_{-k})$ and its variance is the Schur complement ${\Sigma }_{kk}-{\Sigma }_{k,-k}{\Sigma }_{-k,-k}^{-1}{\Sigma }_{-k,k}$. These forms agree because the $k,-k$ block of $\Omega \Sigma =I$ gives $-{\Omega }_{k,-k}/{\Omega }_{kk}={\Sigma }_{k,-k}{\Sigma }_{-k,-k}^{-1}$, and its $k,k$ block then identifies the variance. This proves the relevant conditional multivariate normal distribution formula.

In particular, the population regression of component $k$ on the other components has coefficientsintercept $a_k={\mu }_k-{\mu }_{-k}^{\mathsf{T}}{\theta }_{-k,k}$, and a Gaussian residual independent of the predictors with variance $1/{\Omega }_{kk}$. These are the conditional regression coefficients from a Gaussian precision matrix. The coefficient is zero exactly when the corresponding edge is absent. Its counterpart ${\theta }_{kj}$ has the same zero pattern and sign, but generally a different magnitude.

This motivates nodewise regression: for each $k$, regress $X_k$ on $X_{-k}$, estimating an intercept as well. For sparse graph estimation, use the Nodewise Lasso criterionThe Lasso penalty encourages many fitted coefficients to be zero and remains usable when the number of predictors exceeds the sample size. Ordinary unpenalized sample regressions would usually give a dense estimated graph. Separate regressions can disagree about a pair, so an undirected graph can use the OR rule for nodewise graph selection, retaining an edge if either fitted direction is nonzero, or the more conservative AND rule for nodewise graph selection, retaining it only if both are nonzero. These are choices for combining estimated neighbourhoods, rather than identities forced by finite-sample fits.

The printed joint objective uses a Group Lasso penalty on the unordered pairsThis is paired group Lasso for Gaussian graph estimation. Each group represents a single undirected edge, coupling its two directional regressions. A sum of group norms encourages entire pairs to be zero; a squared group norm would give smooth ridge-type shrinkage instead of edge deletion. The constraint ${\Theta }_{kk}=0$ removes self-prediction, and the intercepts are not penalized.

The free symbols called ${\mu }_k$ in the objective are regression intercepts, not generally the marginal means of the original Gaussian vector. Profiling them out givesand replaces all columns by their centred versions in the squared-error terms. This is profiling intercepts in nodewise regression. Scaling predictors to comparable units can also help interpret a common penalty, although the displayed objective itself is well defined on the given data.

The grouped KKT conditions make the sparsity mechanism precise. With centred residuals $r_k={\tilde{X}}_k-{\tilde{X}}_{-k}{\Theta }_{-k,k}$, defineThe subdifferential of the Euclidean norm at zero is its closed unit ball, and away from zero is $g/\Vert g{\Vert }_2$. Hence a minimizer satisfiesThese are the paired group Lasso optimality conditions. They jointly threshold the two directional scores, matching the population fact that both directions represent the same edge.

Estimate the graph by including $\{j,k\}$ exactly when $\Vert {\hat{g}}_{jk}{\Vert }_2>0$. This graph is undirected by construction. The penalty does not impose ${\Theta }_{jk}={\Theta }_{kj}$, nor does it mathematically require both coordinates of every selected pair to be nonzero. That distinction is appropriate because the population regression coefficients themselves need not have equal magnitudes. The fitted coefficient matrix is not automatically a symmetric positive definite precision matrix; its pair supports, rather than its entries as a precision estimate, supply the requested conditional independence graph.