Triangular matrix

In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.

Because matrix equations with triangular matrices are easier to solve, they are very important in numerical analysis. By the LU decomposition algorithm, an invertible matrix may be written as the product of a lower triangular matrix L and an upper triangular matrix U if and only if all its leading principal minors are non-zero.

Description

A matrix of the form

L

=

[

ℓ

1

,

1

0

⋮

ℓ

2

,

1

ℓ

2

,

2

⋮

ℓ

3

,

1

ℓ

3

,

2

⋱

⋮

⋮

⋱

⋱

ℓ

n

,

1

ℓ

n

,

2

…

ℓ

n

,

n

−

1

ℓ

n

,

n

]

{\displaystyle L={\begin{bmatrix}\ell _{1,1}&&&&0{\vphantom {\vdots }}\\\ell _{2,1}&\ell _{2,2}&&&{\vphantom {\vdots }}\\\ell _{3,1}&\ell _{3,2}&\ddots &&\\\vdots &\vdots &\ddots &\ddots &\\\ell _{n,1}&\ell _{n,2}&\ldots &\ell _{n,n-1}&\ell _{n,n}\end{bmatrix}}}

is called a lower triangular matrix or left triangular matrix, and analogously a matrix of the form

U

=

[

u

1

,

1

u

1

,

2

u

1

,

3

…

u

1

,

n

⋮

u

2

,

2

u

2

,

3

…

u

2

,

n

⋮

⋱

⋱

⋮

⋱

u

n

−

1

,

n

⋮

0

u

n

,

n

⋮

]

{\displaystyle U={\begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots &u_{1,n}{\vphantom {\vdots }}\\&u_{2,2}&u_{2,3}&\ldots &u_{2,n}{\vphantom {\vdots }}\\&&\ddots &\ddots &\vdots \\&&&\ddots &u_{n-1,n}{\vphantom {\vdots }}\\0&&&&u_{n,n}{\vphantom {\vdots }}\end{bmatrix}}}

is called an upper triangular matrix or right triangular matrix. A lower or left triangular matrix is commonly denoted with the variable L, and an upper or right triangular matrix is commonly denoted with the variable U or R.

A matrix that is both upper and lower triangular is diagonal. Matrices that are similar to triangular matrices are called triangularisable.

A non-square (or sometimes any) matrix with zeros above (below) the diagonal is called a lower (upper) trapezoidal matrix. The non-zero entries form the shape of a trapezoid.