15-00-StrictlyUpperTriangularMatrix.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{StrictlyUpperTriangularMatrix}
\pmcreated{2013-03-22 13:42:15}
\pmmodified{2013-03-22 13:42:15}
\pmowner{Daume}{40}
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\pmtitle{strictly upper triangular matrix}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
\pmsynonym{supertriangular}{StrictlyUpperTriangularMatrix}
\pmdefines{strictly lower triangular matrix}

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\begin{document}
A \emph{strictly \PMlinkescapetext{upper triangular matrix}} is an upper triangular matrix which has $0$ on the main diagonal.  Similarly a \emph{strictly lower triangular matrix} is a lower triangular matrix which has $0$ on the main diagonal. i.e.

A strictly upper triangular matrix is of the form

$$ \begin{bmatrix}
0 & a_{12} & a_{13} & \cdots & a_{1n} \\
0 & 0 & a_{23} & \cdots & a_{2n} \\
0 & 0 & 0 & \cdots & a_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 0
\end{bmatrix} $$

A strictly lower triangular matrix is of the form

$$ \begin{bmatrix}
0 & 0 & 0 & \cdots & 0 \\
a_{21} & 0 & 0 & \cdots & 0 \\
a_{31} & a_{32} & 0 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & a_{n3} & \cdots & 0
\end{bmatrix} $$
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