Triangular matrix: Difference between revisions

Revision as of 16:56, 26 August 2004

Revision as of 18:26, 26 August 2004

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A matrix

:<math> \mathbf{L}=

\begin{pmatrixbmatrix}

l_{1,1} & & & & 0 \\

l_{2,1} & l_{2,2} & & & \\

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\vdots & \vdots & \ddots & \ddots & \\

l_{n,1} & l_{n,2} & \ldots & l_{n,n-1} & l_{n,n}

\end{pmatrixbmatrix}

</math>

is called '''lower triangular matrix''' or '''left triangular matrix''', and analogously a matrix of the form

:<math> \mathbf{U} =

\begin{pmatrixbmatrix}

u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\

& u_{2,2} & u_{2,3} & \ldots & u_{2,n} \\

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& & & \ddots & u_{n-1,n}\\

0 & & & & u_{n,n}

\end{pmatrixbmatrix}

</math>

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The matrix

:<math> \mathbf{L}_n =

\begin{pmatrixbmatrix}

1 & & & & & 0 \\

& \ddots & & & & \\

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& & \vdots & & \ddots & \\

0 & & l_{n,n} & & & 1 \\

\end{pmatrixbmatrix}

</math>

is called '''atomic lower triangular''' matrix, with

:<math> \mathbf{U}_n =

\begin{pmatrixbmatrix}

1 & & & l_{1,n} & & 0 \\

& \ddots & & \vdots & & \\

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& & & & \ddots & \\

0 & & & & & 1 \\

\end{pmatrixbmatrix}

</math>

being called '''atomic upper triangular''' matrix.

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== Examples ==

:<math>
\begin{pmatrixbmatrix}

1 & 4 & 2 \\

0 & 3 & 4 \\

0 & 0 & 1 \\

\end{pmatrixbmatrix}
</math>

is upper triangular and

:<math>
\begin{pmatrixbmatrix}

1 & 0 & 0 \\

2 & 8 & 0 \\

4 & 9 & 7 \\

\end{pmatrixbmatrix}
</math>

is lower triangular.