Triangular matrix: Difference between revisions

Revision as of 14:27, 15 August 2004

Revision as of 22:29, 15 August 2004

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== Definition ==

=== Triangular matrix ===

A matrix

:<math> \mathbf{L}=

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

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

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

\end{pmatrix}

</math>

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

Analogously a matrix

:<math> \mathbf{U} =

\begin{pmatrix}

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

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

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

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

0 & & & & u_{Nn,Nn}

\end{pmatrix}

</math>

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is called '''upper triangular matrix''' or '''right triangular matrix'''.

If the entries ''u''_''ii''=1 for all ''i'' &le; ''n'', the matrix is termed '''unit''' upper/lower or '''normed''' upper/lower triangular.

=== Normed triangular matrix ===

A lower triangular matrix where entries in the main diagonal are one

:<math> \mathbf{L} =

\begin{pmatrix}

1 & & & & 0 \\

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

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

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

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

\end{pmatrix}

</math>

is called '''unit lower triangular matrix''' or '''normed lower triangular matrix'''.

Analogously the matrix

:<math> \mathbf{U} =

\begin{pmatrix}

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

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

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

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

0 & & & & 1

\end{pmatrix}

</math>

the matrix is called '''unit upper triangular matrix''' or '''normed upper triangular matrix'''.

=== Atomic triangular matrix ===

The matrix

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

& & \vdots & & \ddots & \\

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

\end{pmatrix}

</math>

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

Analogously the matrix

:<math> \mathbf{U}_n =

\begin{pmatrix}

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\end{pmatrix}

</math>

isbeing called '''atomic upper triangular''' matrix.

== Notes ==