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| Title of object: |
triangular matrix |
| Canonical Name: |
TriangularMatrix |
| Type: |
Definition |
| Created on: |
2002-01-16 07:46:11 |
| Modified on: |
2006-10-21 13:18:07 |
| Classification: |
msc:15-00, msc:65-00 |
| Defines: |
upper triangular, lower triangular, upper triangular matrix, lower triangular matrix, right triangular, right triangular matrix, left triangular, left triangular matrix |
Preamble:
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Content:
\PMlinkescapeword{even}
\section{Triangular Matrix}
A \emph{triangular matrix} is either an \emph{upper triangular matrix} or \emph{lower triangular matrix}.
An upper triangular matrix is of the form
$$ \begin{bmatrix}
a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
0 & a_{22} & a_{23} & \cdots & a_{2n} \\
0 & 0 & a_{33} & \cdots & a_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & a_{nn}
\end{bmatrix} $$
An upper triangular matrix is sometimes also called \emph{right triangular}.
A lower triangular matrix is of the form
$$ \begin{bmatrix}
a_{11} & 0 & 0 & \cdots & 0 \\
a_{21} & a_{22} & 0 & \cdots & 0 \\
a_{31} & a_{32} & a_{33} & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn}
\end{bmatrix} $$
A lower triangular matrix is sometimes also called \emph{left triangular}.
Triangular matrices allow numerous algorithmic shortcuts in many situations. For example, the equation $Ax=b$ can be solved for $x$ in $n^2$ operations, where $A$ is an $n\times n$ triangular matrix.
In fact, triangular matrices are so useful that much computational linear algebra begins with factoring (or decomposing) a general matrix or matrices into triangular form. Some matrix factorization methods are the Cholesky factorization and the LU-factorization. Even including the factorization step, enough later operations are typically avoided to yield an overall time savings.
\section{Properties}
Triangular matrices have the following properties (prefix ``triangular'' with either ``upper'' or ``lower'' uniformly):
\begin{itemize}
\item The inverse of a triangular matrix is a triangular matrix.
\item The product of two triangular matrices is a triangular matrix.
\item The determinant of a triangular matrix is the product of the diagonal elements.
\item The eigenvalues of a triangular matrix are the diagonal elements.
\end{itemize}
The last two properties follow easily from the cofactor expansion of the triangular matrix. |
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