Triangular matrix: Difference between revisions

Revision as of 01:01, 30 March 2005

Revision as of 01:35, 30 March 2005

Line 43:

:<math> \mathbf{U}_{i} =

\begin{bmatrix}

1 & & & l_u_{1,i} & & 0 \\

& \ddots & & \vdots & & \\

& & \ddots & l_u_{i-1,i} & & \\

& & & 1 & & \\

& & & & \ddots & \\

Line 52:

</math>

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

The atomic lower or upper triangular matricies have inverses of the form

:<math> \mathbf{L}_{i}^{-1} =

\begin{bmatrix}

1 & & & & & 0 \\

& \ddots & & & & \\

& & 1 & & & \\

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

& & \vdots & & \ddots & \\

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

\end{bmatrix}

</math>

and

:<math> \mathbf{U}_{i}^{-1} =

\begin{bmatrix}

1 & & & -u_{1,i} & & 0 \\

& \ddots & & \vdots & & \\

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

& & & 1 & & \\

& & & & \ddots & \\

0 & & & & & 1 \\

\end{bmatrix}

</math>

i.e. the elements not on the diagonal are made negative.

== Notes ==

Line 84 ⟶ 108:

</math>

is lower triangular.

Atomic lower triangular matrix inverse

:<math>

\begin{bmatrix}

1 & 0 & 0 \\

4 & 1 & 0 \\

2 & 0 & 1 \\

\end{bmatrix}^{-1} =

\begin{bmatrix}

1 & 0 & 0 \\

-4 & 1 & 0 \\

-2 & 0 & 1 \\

\end{bmatrix}

</math>

== Application ==