Triangular matrix

In linear algebra, 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

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

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

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

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

The matrix L_i

${\displaystyle \mathbf {L} _{i}={\begin{pmatrix}1&&&&&0\\&\ddots &&&&\\&&1&&&\\&&l_{i+1,i}&\ddots &&\\&&\vdots &&\ddots &\\0&&l_{n,i}&0&&1\\\end{pmatrix}}}$ 

is called atomic lower triangular matrix.

Analogously a matrix U of the form

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

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

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

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.

Application

A matrix equation in the form

${\displaystyle \mathbf {L} \mathbf {x} =\mathbf {b} }$ 

or

${\displaystyle \mathbf {U} \mathbf {x} =\mathbf {b} }$ 

is very easy to solve. The matrix equation Lx= b can be written as a system of linear equations

${\displaystyle {\begin{matrix}x_{1}&&&&&=&b_{1}\\l_{2,1}x_{1}&+&x_{2}&&&=&b_{2}\\\vdots &&\vdots &\ddots &&&\vdots \\l_{m,1}x_{1}&+&l_{m,2}x_{2}&+\ldots +&x_{m}&=&b_{m}\\\end{matrix}}}$ 

which can be solved by the following rekursive relation

${\displaystyle {\begin{matrix}x_{1}&=&b_{1}\\x_{2}&=&b_{2}-l_{2,1}b_{1}\\&\vdots &\\x_{m}&=&b_{m}-\sum _{i=1}^{m-1}l_{m,i}x_{i}\end{matrix}}:}$ 

A matrix equation with a normed upper triangular matrix R can be solved in an analogous way.

Because triangular matrices are easy to calculate they are very important in numerical analysis.The LU decomposition gives an algorithm to decompose any invertible matrix A into a normed upper triangle matrix L and a normed lower triangle matrix R.

See also

• Gaussian elimination
• LU decomposition
• QR decomposition
• Hessenberg matrix