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| Title of object: |
triangular matrix |
| Canonical Name: |
TriangularMatrix |
| Type: |
Definition |
| Created on: |
2002-01-16 07:46:11 |
| Modified on: |
2003-03-30 13:05:46 |
| Classification: |
msc:15-00, msc:65-00 |
| Synonyms: |
triangular matrix=upper triangular
triangular matrix=lower triangular
triangular matrix=upper triangular matrix
triangular matrix=lower triangular matrix |
Revision comment (for changes between this and next version):
Changes for correction #3803 ('triangular matrix'):
> 1) could you define 'triangular matrix': a matrix that is either
lower or upper triangular.
Done.
> 2) the terms being defined could be emphasized(?)
Done.
> 3) Maybe a link to the entry on Cholesky decomposition.
Done, and talked about matrix factorizations more.
Other enhancements: fixed some quotes, added sections. |
Preamble:
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Content:
An upper triangular matrix is of the form
$$ \begin{bmatrix}
a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
0 & a_{22} & a_{23} & \cdots & a_{2n} \\
0 & 0 & a_{33} & \cdots & a_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & a_{nn}
\end{bmatrix} $$
A lower triangular matrix is of the form
$$ \begin{bmatrix}
a_{11} & 0 & 0 & \cdots & 0 \\
a_{21} & a_{22} & 0 & \cdots & 0 \\
a_{31} & a_{32} & a_{33} & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn}
\end{bmatrix} $$
Triangular matrices allow numerous algorithmic shortcuts in many situations. For example, $Ax=b$ can be solved in $n^2$ operations if $A$ is an $n\times n$ triangular matrix.
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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