CVE/LaTeX/notes.tex

\documentclass[12pt,a4paper]{article}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{tikz}
\usepackage{fullpage}

\newcommand{\vecl}{\ensuremath{\operatorname{vec}_l}}
\newcommand{\Sym}{\ensuremath{\operatorname{Sym}}}

\begin{document}

Indexing a given matrix $A = (a_{ij})_{i,j = 1, ..., n} \in \mathbb{R}^{n\times n}$ given as
\begin{displaymath}
    A = \begin{pmatrix}
        a_{0,0} & a_{0,1} & a_{0,2} & \ldots & a_{0,n-1} \\
        a_{1,0} & a_{1,1} & a_{1,2} & \ldots & a_{1,n-1} \\
        a_{2,0} & a_{2,1} & a_{2,2} & \ldots & a_{2,n-1} \\
        \vdots & \vdots & \vdots & \ddots & \vdots \\
        a_{n-1,0} & a_{n-1,1} & a_{n-1,2} & \ldots & a_{n-1,n-1}
    \end{pmatrix}
\end{displaymath}

A symmetric matrix with zero main diagonal, meaning a matrix $S = S^T$ with $S_{i,i} = 0,\ \forall i = 1,..,n$ is givne in the following form
\begin{displaymath}
    S = \begin{pmatrix}
        0 & s_{1,0} & s_{2,0} & \ldots & s_{n-1,0} \\
        s_{1,0} & 0 & s_{2,1} & \ldots & s_{n-1,1} \\
        s_{2,0} & s_{2,1} & 0 & \ldots & s_{n-1,2} \\
        \vdots & \vdots & \vdots & \ddots & \vdots \\
        s_{n-1,0} & s_{n-1,1} & s_{n-1,2} & \ldots & 0
    \end{pmatrix}
\end{displaymath}
Therefore its sufficient to store only the lower triangular part, for memory efficiency and some further alrogithmic shortcuts (sometime they are more expencife) the symmetric matrix $S$ is stored in packed form, meanin in a vector of the length $\frac{n(n-1)}{2}$. We use (like for matrices) a column-major order of elements and define the $\vecl:\Sym(n)\to \mathbb{R}^{n(n-1) / 2}$ opperator defined as

\begin{displaymath}
    \vecl(S) = (s_{1,0}, s_{2,0},\cdots,s_{n-1,0},s_{2,1}\cdots,s_{n-1,n-2})^T
\end{displaymath}

The relation between the matrix indices $i,j$ and the $\vecl$ index $k$ is given by

\begin{displaymath}
    (\vecl(S)_k = s_{i,j} \quad\Leftrightarrow\quad k = jn+i) : j \in \{0,...,n-2\} \land j < i < n.
\end{displaymath}

\begin{center}
    \begin{tikzpicture}[xscale=1,yscale=-1]
        % \foreach \i in {0,...,5} {
        %     \node at ({mod(\i, 3)}, {int(\i / 3)}) {$\i$};
        % }
        \foreach \i in {1,...,4} {
            \foreach \j in {1,...,\i} {
                \node at (\j, \i) {$\i,\j$};
            }
        }
        
    \end{tikzpicture}
\end{center}

\end{document}