Triangular matrix: Difference between revisions

Revision as of 14:23, 15 August 2004

Revision as of 14:27, 15 August 2004

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In the [[mathematics | mathematical]] discipline of [[linear algebra]], a '''triangular matrix''' is a special kind of [[Matrix_(mathematics)|square matrix]] where the entries below or above the [[main diagonal]] are zero.

Because [[Matrix_(mathematics)|matrix]] equations with triangular matrices are easy to solve they are very important in [[numerical analysis]]. The [[LU decomposition]] gives an algorithm to decompose any [[invertible matrix]] ''A'' into a normed lower triangle matrix ''L'' and an upper triangle matrix ''U''.

== Definition ==

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l_{3,1} & l_{3,2} & \ddots & & \\

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

l_{nN,1} & l_{nN,2} & \ldots & l_{nN,mN-1} & l_{nN,mN}

\end{pmatrix}

</math>

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:<math> \mathbf{U} =

\begin{pmatrix}

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

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

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

& & & \ddots & u_{nN-1,mN}\\

0 & & & & u_{nN,mN}

\end{pmatrix}

</math>

Line 42:

l_{3,1} & l_{3,2} & \ddots & & \\

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

l_{nN,1} & l_{nN,2} & \ldots & l_{nN,mN-1} & 1

\end{pmatrix}

</math>

Line 52:

:<math> \mathbf{U} =

\begin{pmatrix}

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

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

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

& & & \ddots & u_{nN-1,mN}\\

0 & & & & 1

\end{pmatrix}

Line 64:

The matrix

:<math> \mathbf{L}_i_n =

\begin{pmatrix}

1 & & & & & 0 \\

& \ddots & & & & \\

& & 1 & & & \\

& & l_{in+1,in} & \ddots & & \\

& & \vdots & & \ddots & \\

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

\end{pmatrix}

</math>

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Analogously the matrix

:<math> \mathbf{U}_i_n =

\begin{pmatrix}

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

& \ddots & & \vdots & & \\

& & \ddots & l_{in-1,in} & & \\

& & & 1 & & \\

& & & & \ddots & \\