Triangular matrix

In mathematics, a triangular matrix is a matrix with entries that have elements either completely above (upper triangular) or completely below (lower triangular) the principal diagonal.

For example:

${\displaystyle {\begin{pmatrix}1&4&2\\0&3&4\\0&0&1\\\end{pmatrix}}}$

is upper triangular and

${\displaystyle {\begin{pmatrix}1&0&0\\2&8&0\\4&9&7\\\end{pmatrix}}}$

is lower triangular.

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 L is usually unit lower triangular.

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

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, QR decomposition, Hessenberg matrix

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