Triangular matrix: Difference between revisions

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[[Category:Linear algebra]]

In [[mathematicsmatrix theory]], a '''triangular matrix''' is a special kind of [[Matrix_(mathematics)|matrix]] withwhere the entries thatbelow have elements either completelyor above ('''upper triangular''') or completely below ('''lower triangular''') the [[principalmain diagonal]] are zero.

== Definition ==

A matrix ''L'' of the form

:<math>

\begin{pmatrix}

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

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

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

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

l_{n,1} & l_{n,2} & \ldots & l_{n,m-1} & l_{n,m}

\end{pmatrix}

</math>

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

:<math>

\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,m-1} & 1

\end{pmatrix}

</math>

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

Analogously a matrix ''U'' of the form

:<math>

\begin{pmatrix}

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

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

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

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

0 & & & & u_{n,m}

\end{pmatrix}

</math>

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

:<math>

\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 ==

The [[identity matrix]] is a 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 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.'

Generally, operations can be performed on triangular matrices within half the time.

== Examples ==

For example:

:<math>\begin{pmatrix}

1 & 4 & 2 \\

Line 12 ⟶ 75:

\end{pmatrix}</math>

is upper triangular and

:<math>\begin{pmatrix}

1 & 0 & 0 \\

Line 21 ⟶ 85:

It is also sometimes useful to distinguish matrices that are '''unit lower triangular''' or '''unit upper triangular'''. These matrices are triangular with the additional property that all of the diagonal entries are 1. In [[LU decomposition]], the matrix <var>L</var> is usually unit lower triangular.

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.

== See also ==

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.'

Generally, operations can be performed on triangular matrices within half the time.

==See also==

* [[Row echelon form]]

* [[LU decomposition]]