Triangular matrix: Difference between revisions

Revision as of 01:35, 30 March 2005

Revision as of 15:38, 30 March 2005

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

If the entries on the [[principal diagonal]] are 1, the matrix is termed '''unit''' upper/lower or '''normed''' upper/lower triangular. If, in addition, all the off-diagonal entries are zero except for the entries in one column, then the matrix is '''atomic''' upper/lower triangular. So an atomic lower triangular matrix is of the form

The matrix

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

\begin{bmatrix}

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& & \vdots & & \ddots & \\

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

\end{bmatrix}.

</math>

isThe calledinverse of an '''atomic lowertriagular matrix is again atomic triangular'''. matrixIndeed, withwe have

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

\begin{bmatrix}

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

& \ddots & & \vdots & & \\

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

& & & 1 & & \\

& & & & \ddots & \\

0 & & & & & 1 \\

\end{bmatrix}

</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}

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& & \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 elementsoff-diagonal notentries onare thereplaced diagonal areby madetheir negativeopposites.

== Notes ==

A matrix which is simultaneously upper and lower triangular is [[diagonal matrix|diagonal]]. The [[identity matrix]] is athe only matrix which is both normed upper and lower triangular matrix.

The product of two upper triangular matrices is upper triangular, so the set of upper triangular matrices forms an [[associative algebra|algebra]]. Algebras of upper triangular matrices have a natural generalisation in [[functional analysis]] which yields [[nest algebra]]s.

The [[transpose]] of a upper triangular matrix is a lower triangular matrix and vice versa. The [[determinant]] of a triangular matrix equals the product of the diagonal entries.

The variable L is commonly used for lower triangular matrix, standing for lower/left, while the variable U or R is commonly used for upper triangular matrix, standing for upper/right. The variable R has the added benefit of being the same initial for the German term for 'right.'

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== Examples ==

The matrix

:<math>

\begin{bmatrix}

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</math>

is upper triangular and

:<math>

\begin{bmatrix}

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is lower triangular.

The matrix

Atomic lower triangular matrix inverse

:<math>

\begin{bmatrix}

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4 & 1 & 0 \\

2 & 0 & 1 \\

\end{bmatrix}^{-1} =

</math>

Atomicis atomic lower triangular matrixand its inverse is

:</math>

\begin{bmatrix}

1 & 0 & 0 \\

-4 & 1 & 0 \\

-2 & 0 & 1 \\

\end{bmatrix}.

</math>