Triangular matrix

In matrix theory, a triangular matrix is a special kind of matrix where the entries below or above the main diagonal are zero.

Definition

A matrix L of the form

${\displaystyle {\begin{pmatrix}l_{1,1}&&&&0\\l_{2,1}&l_{2,2}&&&\\l_{3,1}&l_{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\l_{n,1}&l_{n,2}&\ldots &l_{n,m-1}&l_{n,m}\end{pmatrix}}}$ 

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

${\displaystyle {\begin{pmatrix}1&&&&0\\l_{2,1}&1&&&\\l_{3,1}&l_{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\l_{n,1}&l_{n,2}&\ldots &l_{n,m-1}&1\end{pmatrix}}}$ 

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

Analogously a matrix U of the form

${\displaystyle {\begin{pmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots &u_{1,m}\\&u_{2,2}&u_{2,3}&\ldots &u_{2,m}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,m}\\0&&&&u_{n,m}\end{pmatrix}}}$ 

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

${\displaystyle {\begin{pmatrix}1&u_{1,2}&u_{1,3}&\ldots &u_{1,m}\\&1&u_{2,3}&\ldots &u_{2,m}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,m}\\0&&&&1\end{pmatrix}}}$ 

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 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.

Examples

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

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.

See also

• Row echelon form
• LU decomposition
• QR decomposition
• Hessenberg matrix