Triangular matrix: Difference between revisions

Revision as of 14:15, 15 August 2004

Revision as of 14:23, 15 August 2004

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== Definition ==

A=== Triangular matrix ''L'' of the form===

A matrix

:<math> \mathbf{L}=

\begin{pmatrix}

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</math>

is called '''lower triangular matrix''' or '''left triangular matrix'''. If the diagonal entries in ''L'' are one

Analogously a matrix

:<math> \mathbf{U} =

\begin{pmatrix}

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

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

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

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

0 & & & & 1u_{n,m}

\end{pmatrix}

</math>

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

=== Normed triangular matrix ===

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

:<math> \mathbf{L} =

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</math>

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

Analogously the matrix

The matrix ''L''_i

:<math> \mathbf{U} =

\begin{pmatrix}

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

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

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

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

0 & & & & 1

\end{pmatrix}

</math>

the matrix is called '''unit upper triangular matrix''' or '''rightnormed upper triangular matrix'''. If the diagonal entries in ''U'' are one

=== Atomic triangular matrix ===

The matrix

:<math> \mathbf{L}_i =

\begin{pmatrix}

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

& & \vdots & & \ddots & \\

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

\end{pmatrix}

</math>

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

Analogously athe matrix ''U'' of the form

:<math> \mathbf{U}_i =

\begin{pmatrix}

u_{1,1} & u_{ 1,2} & u_{1,3} & \ldots & u_ l_{1,mi} & & 0 \\

& u_{2,2}\ddots & u_{2,3} & \ldotsvdots & & u_{2,m} \\

& & \ddots & \ddotsl_{i-1,i} & \vdots & \\

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

0 & & & & u_{n,m} \ddots & \\

0 & & & & & 1 \\

\end{pmatrix}

</math>

is called '''atomic upper triangular''' matrix.

is called '''upper triangular matrix''' or '''right triangular matrix'''. If the diagonal entries in ''U'' are one

:<math> \mathbf{U} =

\begin{pmatrix}

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

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

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

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

0 & & & & 1

\end{pmatrix}

</math>

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

== Notes ==