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Inverses/products of triangular matrices |
Tom Lougheed (talk | contribs) Removed false claim |
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The article clearly states that products of upper triangular matrices are upper triangular, but it doesn't make the similar (and also true) claim about lower triangular matrices. Further, I only vaguely get the impression that the inverses of upper/lower triangular matrices remain upper/lower triangular. We should probably state these properties more directly, and perhaps clean up the article in general. --[[User:Rriegs|Rriegs]] 04:11, 5 May 2007 (UTC)
== Removed false claim ==
I commented out a part of the article falsely claimed that
:''"Indeed, we have''
::<math> \mathbf{L}^{-1} =
\begin{bmatrix}
1 & & & & & 0 \\
& \ddots & & & & \\
& & 1 & & & \\
& &-l_{i+1,i} & \ddots & & \\
& & \vdots & & \ddots & \\
0 & & -l_{n,i} & & & 1 \\
\end{bmatrix},
</math>
:''i.e. the off-diagonal entries are replaced by their opposites."''
Except for the first sub-diagonal, the inverse of an atomic lower triangluar is '''not''' quite as simple as reversing signs.
Consider this counter example:
:<math> \mathbf{L} =
\begin{bmatrix}
1 & & \\
2 & 1 & \\
3 & 4 & 1 \\
\end{bmatrix}
</math>
Notice that
<math>
\begin{bmatrix}
1 & & \\
2 & 1 & \\
3 & 4 & 1 \\
\end{bmatrix} \begin{bmatrix}
1 & & \\
-2 & 1 & \\
-3 & -4 & 1 \\
\end{bmatrix} = \begin{bmatrix}
1 & & \\
0 & 1 & \\
-8 & 0 & 1 \\
\end{bmatrix} \ne \mathbf{I}
</math>
[[User:Tom Lougheed|Tom Lougheed]] 01:17, 12 August 2007 (UTC)
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