46 lines
2.0 KiB
TeX
46 lines
2.0 KiB
TeX
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\documentclass[12pt,a4paper]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{amsmath, amsfonts, amssymb, amsthm}
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\usepackage{fullpage}
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\newcommand{\vecl}{\ensuremath{\operatorname{vec}_l}}
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\newcommand{\Sym}{\ensuremath{\operatorname{Sym}}}
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\begin{document}
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Indexing a given matrix $A = (a_{ij})_{i,j = 1, ..., n} \in \mathbb{R}^{n\times n}$ given as
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\begin{displaymath}
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A = \begin{pmatrix}
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a_{0,0} & a_{0,1} & a_{0,2} & \ldots & a_{0,n-1} \\
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a_{1,0} & a_{1,1} & a_{1,2} & \ldots & a_{1,n-1} \\
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a_{2,0} & a_{2,1} & a_{2,2} & \ldots & a_{2,n-1} \\
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\vdots & \vdots & \vdots & \ddots & \vdots \\
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a_{n-1,0} & a_{n-1,1} & a_{n-1,2} & \ldots & a_{n-1,n-1}
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\end{pmatrix}
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\end{displaymath}
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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
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\begin{displaymath}
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S = \begin{pmatrix}
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0 & s_{1,0} & s_{2,0} & \ldots & s_{n-1,0} \\
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s_{1,0} & 0 & s_{2,1} & \ldots & s_{n-1,1} \\
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s_{2,0} & s_{2,1} & 0 & \ldots & s_{n-1,2} \\
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\vdots & \vdots & \vdots & \ddots & \vdots \\
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s_{n-1,0} & s_{n-1,1} & s_{n-1,2} & \ldots & 0
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\end{pmatrix}
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\end{displaymath}
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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
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\begin{displaymath}
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\vecl(S) = (s_{1,0}, s_{2,0},\cdots,s_{n-1,0},s_{2,1}\cdots,s_{n-1,n-2})^T
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\end{displaymath}
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The relation between the matrix indices $i,j$ and the $\vecl$ index $k$ is given by
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\begin{displaymath}
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(\vecl(S)_k = s_{i,j} \quad\Leftrightarrow\quad k = jn+i) : j \in \{0,...,n-2\} \land j < i < n.
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\end{displaymath}
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\end{document}
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