From aac70fbd5300b1edb4ffb14e64eee0dc5913993b Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:03:18 +0100 Subject: [PATCH 1/8] fix rendering issues --- docs/1_fixed_effects_and_block_methods.md | 35 ++++++++++++----------- docs/3_local_solvers.md | 1 + 2 files changed, 19 insertions(+), 17 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 6c31f0c..9159152 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -37,10 +37,11 @@ $$ where $f_q(i) \in \{1, \dots, m_q\}$ is the level of factor $q$ for observation $i$ and $\alpha_{q,j}$ is the coefficient for level $j$ of factor $q$. -In matrix form, we could writ this as +In matrix form, we could write this as -$$y = D\alpha + \varepsilon -$$, +$$ +y = D\alpha + \varepsilon +$$ where $D$ is the sparse $n \times m$ design matrix of 0s and 1s. Each row of $D$ has exactly $Q$ ones, one in each factor's column block. @@ -74,11 +75,11 @@ The $m \times m$ Gramian inherits a natural block structure from the factor part $$ G = \begin{pmatrix} -{\color{royalblue}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\ -C_{12}^\top & {\color{crimson}D_2} & C_{23} & \cdots & C_{2Q} \\ -C_{13}^\top & C_{23}^\top & {\color{forestgreen}D_3} & \cdots & C_{3Q} \\ +{\color{#4169E1}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\ +C_{12}^\top & {\color{#DC143C}D_2} & C_{23} & \cdots & C_{2Q} \\ +C_{13}^\top & C_{23}^\top & {\color{#228B22}D_3} & \cdots & C_{3Q} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{goldenrod}D_Q} +C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{#DAA520}D_Q} \end{pmatrix} $$ @@ -109,20 +110,20 @@ The Gramian has $Q = 3$ diagonal blocks and $\binom{3}{2} = 3$ cross-tabulation $$ G = \begin{pmatrix} -{\color{royalblue}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\ -{\color{gray}C_{WF}^\top} & {\color{crimson}D_F} & {\color{gray}C_{FY}} \\ -{\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{forestgreen}D_Y} +{\color{#4169E1}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\ +{\color{gray}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{gray}C_{FY}} \\ +{\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{#228B22}D_Y} \end{pmatrix} = \left(\begin{array}{ccc|cc|cc} -{\color{royalblue}2} & {\color{royalblue}0} & {\color{royalblue}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ -{\color{royalblue}0} & {\color{royalblue}2} & {\color{royalblue}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ -{\color{royalblue}0} & {\color{royalblue}0} & {\color{royalblue}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ +{\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ +{\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ +{\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ \hline -{\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{crimson}3} & {\color{crimson}0} & {\color{gray}2} & {\color{gray}1} \\ -{\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{crimson}0} & {\color{crimson}3} & {\color{gray}1} & {\color{gray}2} \\ +{\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{#DC143C}3} & {\color{#DC143C}0} & {\color{gray}2} & {\color{gray}1} \\ +{\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{#DC143C}0} & {\color{#DC143C}3} & {\color{gray}1} & {\color{gray}2} \\ \hline -{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{forestgreen}3} & {\color{forestgreen}0} \\ -{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{forestgreen}0} & {\color{forestgreen}3} +{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ +{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{#228B22}0} & {\color{#228B22}3} \end{array}\right) $$ diff --git a/docs/3_local_solvers.md b/docs/3_local_solvers.md index 71755f4..230989a 100644 --- a/docs/3_local_solvers.md +++ b/docs/3_local_solvers.md @@ -96,6 +96,7 @@ Each eliminated worker with $d$ firm connections creates a dense **clique** of $ The **approximate** variant (Gao, Kyng, and Spielman, 2025) replaces each clique with a random **spanning tree** with only $d - 1$ edges instead of $\binom{d}{2}$. The tree weights are chosen so that the expected Laplacian matches the clique Laplacian (i.e. is an **unbiased estimator**). For a worker observed at 100 firms, this reduces the fill from 4,950 edges to just 99 - a 50× reduction - without introducing bias, since the tree weights are chosen so that the approximate Schur complement is unbiased. + --- ## 4. Approximate Cholesky Factorization From 2364e402666744c9365b5c3db1265d9d0f55fe69 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:24:28 +0100 Subject: [PATCH 2/8] fix rendering issues --- docs/1_fixed_effects_and_block_methods.md | 8 ++++---- docs/2_solver_architecture.md | 2 +- 2 files changed, 5 insertions(+), 5 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 9159152..194bce1 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -73,7 +73,7 @@ The Gramian $G$ is symmetric positive semi-definite and always singular: within The $m \times m$ Gramian inherits a natural block structure from the factor partition. With columns ordered by factor ($m_1$ columns for factor 1, then $m_2$ for factor 2, etc.): -$$ +```math G = \begin{pmatrix} {\color{#4169E1}D_1} & C_{12} & C_{13} & \cdots & C_{1Q} \\ C_{12}^\top & {\color{#DC143C}D_2} & C_{23} & \cdots & C_{2Q} \\ @@ -81,7 +81,7 @@ C_{13}^\top & C_{23}^\top & {\color{#228B22}D_3} & \cdots & C_{3Q} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ C_{1Q}^\top & C_{2Q}^\top & C_{3Q}^\top & \cdots & {\color{#DAA520}D_Q} \end{pmatrix} -$$ +``` The blocks are: @@ -108,7 +108,7 @@ Factor 1 (the worker fixed effect) has $m_1 = 3$ levels: {W1, W2, W3}. Factor 2 The Gramian has $Q = 3$ diagonal blocks and $\binom{3}{2} = 3$ cross-tabulation blocks: -$$ +```math G = \begin{pmatrix} {\color{#4169E1}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\ {\color{gray}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{gray}C_{FY}} \\ @@ -125,7 +125,7 @@ G = \begin{pmatrix} {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{#228B22}0} & {\color{#228B22}3} \end{array}\right) -$$ +``` $D_W$ is $3 \times 3$ (one row/column per worker) with 2s on the diagonal because each worker appears in exactly 2 observations (e.g. W1 in obs 1, 2). Off-diagonals are zero because no observation belongs to two workers. $D_F$ is $2 \times 2$ with 3s on the diagonal because each firm appears in 3 observations (F1 in obs 1, 3, 4; F2 in obs 2, 5, 6). The cross-tabulation block $C_{WY}$ is $3 \times 2$ (3 workers $\times$ 2 years); entry $[j,k]$ counts observations where worker $j$ is observed in year $k$. Here every worker appears once per year, so $C_{WY}$ is all ones. diff --git a/docs/2_solver_architecture.md b/docs/2_solver_architecture.md index 619ebea..556c2f9 100644 --- a/docs/2_solver_architecture.md +++ b/docs/2_solver_architecture.md @@ -37,7 +37,7 @@ As discussed in [Part 1, Section 5.2](1_fixed_effects_and_block_methods.md#52-th ## 2. Graph Structure of the Gramian -Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $G = \mathcal{D} + \mathcal{C}$, where $\mathcal{D} = \operatorname{block-diag}(D_1, \ldots, D_Q)$ collects the diagonal blocks and $\mathcal{C}$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition. +Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $`G = \mathcal{D} + \mathcal{C}`$, where $`\mathcal{D} = \operatorname{block-diag}(D_1, \ldots, D_Q)`$ collects the diagonal blocks and $`\mathcal{C}`$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition. ### 2.1 Factor-pair bipartite blocks From 674c20dbbf164e0c5e6623810bd9cc9fe4c259a0 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:31:49 +0100 Subject: [PATCH 3/8] another attempt --- docs/1_fixed_effects_and_block_methods.md | 2 -- docs/2_solver_architecture.md | 2 +- 2 files changed, 1 insertion(+), 3 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 194bce1..6c120c3 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -118,10 +118,8 @@ G = \begin{pmatrix} {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ -\hline {\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{#DC143C}3} & {\color{#DC143C}0} & {\color{gray}2} & {\color{gray}1} \\ {\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{#DC143C}0} & {\color{#DC143C}3} & {\color{gray}1} & {\color{gray}2} \\ -\hline {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{#228B22}0} & {\color{#228B22}3} \end{array}\right) diff --git a/docs/2_solver_architecture.md b/docs/2_solver_architecture.md index 556c2f9..fad1039 100644 --- a/docs/2_solver_architecture.md +++ b/docs/2_solver_architecture.md @@ -37,7 +37,7 @@ As discussed in [Part 1, Section 5.2](1_fixed_effects_and_block_methods.md#52-th ## 2. Graph Structure of the Gramian -Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $`G = \mathcal{D} + \mathcal{C}`$, where $`\mathcal{D} = \operatorname{block-diag}(D_1, \ldots, D_Q)`$ collects the diagonal blocks and $`\mathcal{C}`$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition. +Part 1 derived the block structure of $G = D^\top W D$, with diagonal blocks $D_q$ and cross-tabulation blocks $C_{qr}$. It is convenient to write $`G = \mathcal{D} + \mathcal{C}`$, where $`\mathcal{D} = \text{block-diag}(D_1, \ldots, D_Q)`$ collects the diagonal blocks and $`\mathcal{C}`$ collects the off-diagonal cross-tabulation blocks. This section describes the graph-theoretic properties that drive the domain decomposition. ### 2.1 Factor-pair bipartite blocks From b3ab5b3c536028ea57145b60fcb9455bc79e5f91 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:35:17 +0100 Subject: [PATCH 4/8] another attempt --- docs/1_fixed_effects_and_block_methods.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 6c120c3..125a250 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -114,7 +114,7 @@ G = \begin{pmatrix} {\color{gray}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{gray}C_{FY}} \\ {\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{#228B22}D_Y} \end{pmatrix} -= \left(\begin{array}{ccc|cc|cc} += \begin{pmatrix} {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ @@ -122,7 +122,7 @@ G = \begin{pmatrix} {\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{#DC143C}0} & {\color{#DC143C}3} & {\color{gray}1} & {\color{gray}2} \\ {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{#228B22}0} & {\color{#228B22}3} -\end{array}\right) +\end{pmatrix} ``` $D_W$ is $3 \times 3$ (one row/column per worker) with 2s on the diagonal because each worker appears in exactly 2 observations (e.g. W1 in obs 1, 2). Off-diagonals are zero because no observation belongs to two workers. $D_F$ is $2 \times 2$ with 3s on the diagonal because each firm appears in 3 observations (F1 in obs 1, 3, 4; F2 in obs 2, 5, 6). The cross-tabulation block $C_{WY}$ is $3 \times 2$ (3 workers $\times$ 2 years); entry $[j,k]$ counts observations where worker $j$ is observed in year $k$. Here every worker appears once per year, so $C_{WY}$ is all ones. From 02fd8bad4e220085feba21e3e886513807a41116 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:37:19 +0100 Subject: [PATCH 5/8] another attempt --- docs/1_fixed_effects_and_block_methods.md | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 125a250..0482c35 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -114,7 +114,10 @@ G = \begin{pmatrix} {\color{gray}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{gray}C_{FY}} \\ {\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{#228B22}D_Y} \end{pmatrix} -= \begin{pmatrix} +``` + +```math +G = \begin{pmatrix} {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ From b34caac355eb77eeb346aebb7896075053aaa785 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:39:06 +0100 Subject: [PATCH 6/8] another attempt --- docs/1_fixed_effects_and_block_methods.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 0482c35..407fcf4 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -118,13 +118,13 @@ G = \begin{pmatrix} ```math G = \begin{pmatrix} -{\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} \\ -{\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{gray}2} & {\color{gray}0} & {\color{gray}1} & {\color{gray}1} \\ -{\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{gray}0} & {\color{gray}2} & {\color{gray}1} & {\color{gray}1} \\ -{\color{gray}1} & {\color{gray}2} & {\color{gray}0} & {\color{#DC143C}3} & {\color{#DC143C}0} & {\color{gray}2} & {\color{gray}1} \\ -{\color{gray}1} & {\color{gray}0} & {\color{gray}2} & {\color{#DC143C}0} & {\color{#DC143C}3} & {\color{gray}1} & {\color{gray}2} \\ -{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{gray}1} & {\color{#228B22}3} & {\color{#228B22}0} \\ -{\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}1} & {\color{gray}2} & {\color{#228B22}0} & {\color{#228B22}3} +2 & 0 & 0 & 1 & 1 & 1 & 1 \\ +0 & 2 & 0 & 2 & 0 & 1 & 1 \\ +0 & 0 & 2 & 0 & 2 & 1 & 1 \\ +1 & 2 & 0 & 3 & 0 & 2 & 1 \\ +1 & 0 & 2 & 0 & 3 & 1 & 2 \\ +1 & 1 & 1 & 2 & 1 & 3 & 0 \\ +1 & 1 & 1 & 1 & 2 & 0 & 3 \end{pmatrix} ``` From 63181bb781dbefea53e5b3a65a71f1d218418d62 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:40:19 +0100 Subject: [PATCH 7/8] another attempt --- docs/1_fixed_effects_and_block_methods.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 407fcf4..654b2bc 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -118,13 +118,13 @@ G = \begin{pmatrix} ```math G = \begin{pmatrix} -2 & 0 & 0 & 1 & 1 & 1 & 1 \\ -0 & 2 & 0 & 2 & 0 & 1 & 1 \\ -0 & 0 & 2 & 0 & 2 & 1 & 1 \\ -1 & 2 & 0 & 3 & 0 & 2 & 1 \\ -1 & 0 & 2 & 0 & 3 & 1 & 2 \\ -1 & 1 & 1 & 2 & 1 & 3 & 0 \\ -1 & 1 & 1 & 1 & 2 & 0 & 3 +{\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & 1 & 1 & 1 & 1 \\ +{\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & 2 & 0 & 1 & 1 \\ +{\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & 0 & 2 & 1 & 1 \\ +1 & 2 & 0 & {\color{#DC143C}3} & {\color{#DC143C}0} & 2 & 1 \\ +1 & 0 & 2 & {\color{#DC143C}0} & {\color{#DC143C}3} & 1 & 2 \\ +1 & 1 & 1 & 2 & 1 & {\color{#228B22}3} & {\color{#228B22}0} \\ +1 & 1 & 1 & 1 & 2 & {\color{#228B22}0} & {\color{#228B22}3} \end{pmatrix} ``` From 943904a869df0d897ccc16893d7dfc6fa75d0c15 Mon Sep 17 00:00:00 2001 From: Alexander Fischer Date: Mon, 23 Mar 2026 23:41:22 +0100 Subject: [PATCH 8/8] another attempt --- docs/1_fixed_effects_and_block_methods.md | 6 +----- 1 file changed, 1 insertion(+), 5 deletions(-) diff --git a/docs/1_fixed_effects_and_block_methods.md b/docs/1_fixed_effects_and_block_methods.md index 654b2bc..ec7cfd3 100644 --- a/docs/1_fixed_effects_and_block_methods.md +++ b/docs/1_fixed_effects_and_block_methods.md @@ -113,11 +113,7 @@ G = \begin{pmatrix} {\color{#4169E1}D_W} & {\color{gray}C_{WF}} & {\color{gray}C_{WY}} \\ {\color{gray}C_{WF}^\top} & {\color{#DC143C}D_F} & {\color{gray}C_{FY}} \\ {\color{gray}C_{WY}^\top} & {\color{gray}C_{FY}^\top} & {\color{#228B22}D_Y} -\end{pmatrix} -``` - -```math -G = \begin{pmatrix} +\end{pmatrix} = \begin{pmatrix} {\color{#4169E1}2} & {\color{#4169E1}0} & {\color{#4169E1}0} & 1 & 1 & 1 & 1 \\ {\color{#4169E1}0} & {\color{#4169E1}2} & {\color{#4169E1}0} & 2 & 0 & 1 & 1 \\ {\color{#4169E1}0} & {\color{#4169E1}0} & {\color{#4169E1}2} & 0 & 2 & 1 & 1 \\