Triangular matrix: Difference between revisions

Revision as of 14:43, 29 June 2004

Revision as of 15:30, 29 June 2004

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

== Definition ==

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\end{pmatrix}</math>

is lower triangular.

== Application ==

A matrix equation in the form

:<math>\mathbf{L}\mathbf{x} = \mathbf{b}</math>

or

:<math>\mathbf{U} \mathbf{x} = \mathbf{b}</math>

is very easy to solve. The matrix equation ''Lx''= ''b'' can be written as a system of linear equations

:<math>

\begin{matrix}

x_1 & & & & & = & b_1 \\

l_{2,1} x_1 & + & x_2 & & & = & b_2 \\

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

l_{m,1} x_1 & + & l_{m,2} x_2 & + \ldots + & x_m & = & b_m \\

\end{matrix}

</math>

which can be solved by the following rekursive relation

:<math>

\begin{matrix}

x_1 & = & b_1 \\

x_2 & = & b_2 - l_{2,1} b_1 \\

& \vdots & \\

x_m & = & b_m - \sum_{i=1}^{m-1} l_{m,i}x_i

\end{matrix}

:</math>

A matrix equation with a normed upper triangular matrix R can be solved in an analogous way.

Because triangular matrices are easy to calculate they are very important in numerical analysis.The [[LU decomposition]] gives an algorithm to decompose any [[invertible matrix]] ''A'' into a normed upper triangle matrix ''L'' and a normed lower triangle matrix ''R''.