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15-00-UnitTriangularMatrix.tex
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15-00-UnitTriangularMatrix.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{UnitTriangularMatrix}
\pmcreated{2013-03-22 13:41:58}
\pmmodified{2013-03-22 13:41:58}
\pmowner{Daume}{40}
\pmmodifier{Daume}{40}
\pmtitle{unit triangular matrix}
\pmrecord{6}{34375}
\pmprivacy{1}
\pmauthor{Daume}{40}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
\pmsynonym{unit left triangular matrix}{UnitTriangularMatrix}
\pmsynonym{unit right triangular matrix}{UnitTriangularMatrix}
\pmdefines{unit upper triangular matrix}
\pmdefines{unit lower triangular matrix}
\endmetadata
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\begin{document}
A \emph{unit triangular matrix} is a triangular matrix with $1$ on the diagonal.\\i.e.
A \emph{unit upper triangular matrix} is of the form
$$ \begin{bmatrix}
1 & a_{12} & a_{13} & \cdots & a_{1n} \\
0 & 1 & a_{23} & \cdots & a_{2n} \\
0 & 0 & 1 & \cdots & a_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{bmatrix} $$
and is sometimes called a \emph{unit right triangular matrix}.
A \emph{unit lower triangular matrix} is of the form
$$ \begin{bmatrix}
1 & 0 & 0 & \cdots & 0 \\
a_{21} & 1 & 0 & \cdots & 0 \\
a_{31} & a_{32} & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & a_{n3} & \cdots & 1
\end{bmatrix} $$
and is sometimes called a \emph{unit left triangular matrix}.
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\end{document}