triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. Because matrix equations with triangular matrices are easy to solve they are very important in numerical analysis. The LU decomposition gives an algorithm to decompose any invertible matrix A into a normed lower triangle matrix L and an upper triangle matrix U.

Definition

A matrix

\mathbf{L}=

\begin{bmatrix} 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,n-1} & l_{n,n} \end{bmatrix} is called lower triangular matrix or left triangular matrix, and analogously a matrix of the form

\mathbf{U} =

\begin{bmatrix} u_{1,1} & u_{1,2} & u_{1,3} & \ldots & u_{1,n} \\

         & u_{2,2} & u_{2,3} & \ldots & u_{2,n}  \\         &         & \ddots  & \ddots & \vdots   \\         &         &         & \ddots & u_{n-1,n}\\   0     &         &         &        & u_{n,n}

\end{bmatrix} is called upper triangular matrix or right triangular matrix. If the entries on the principal diagonal are 1, the matrix is termed unit upper/lower or normed upper/lower triangular. If, in addition, all the off-diagonal entries are zero except for the entries in one column, then the matrix is atomic upper/lower triangular; such a matrix is also called a Gauss (transformation) matrix. So an atomic lower triangular matrix is of the form

\mathbf{L}_{i} =

\begin{bmatrix}

      1 &        &           &         &         & 0 \\        & \ddots &           &         &         &   \\        &        &         1 &         &         &   \\        &        & l_{i+1,i} &  \ddots &         &   \\        &        &    \vdots &         &  \ddots &   \\      0 &        &   l_{n,i} &         &         & 1 \\

\end{bmatrix}. The inverse of an atomic triagular matrix is again atomic triangular. Indeed, we have

\mathbf{L}_{i}^{-1} =

\begin{bmatrix}

      1 &        &           &         &         & 0 \\        & \ddots &           &         &         &   \\        &        &         1 &         &         &   \\        &        &-l_{i+1,i} &  \ddots &         &   \\        &        &    \vdots &         &  \ddots &   \\      0 &        &  -l_{n,i} &         &         & 1 \\

\end{bmatrix}, i.e. the off-diagonal entries are replaced by their opposites.

Notes

A matrix which is simultaneously upper and lower triangular is diagonal. The identity matrix is the only matrix which is both normed upper and 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 transpose of a upper triangular matrix is a lower triangular matrix and vice versa. The determinant of a triangular matrix equals the product of the diagonal entries. 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

The matrix

\begin{bmatrix} 1 & 4 & 2 \\ 0 & 3 & 4 \\ 0 & 0 & 1 \\ \end{bmatrix} is upper triangular and

\begin{bmatrix} 1 & 0 & 0 \\ 2 & 8 & 0 \\ 4 & 9 & 7 \\ \end{bmatrix} is lower triangular. The matrix

\begin{bmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 2 & 0 & 1 \\ \end{bmatrix} is atomic lower triangular and its inverse is

\begin{bmatrix}

  1 & 0 & 0 \\

-4 & 1 & 0 \\ -2 & 0 & 1 \\ \end{bmatrix}.

Application

A matrix equation in the form

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

\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

\begin{matrix} l_{1,1} x_1 & & & & & = & b_1 \\ l_{2,1} x_1 & + & l_{2,2} x_2 & & & = & b_2 \\

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

l_{m,1} x_1 & + & l_{m,2} x_2 & + \ldots + & l_{m,m} x_m & = & b_m \\ \end{matrix} which can be solved by the following recursive relation

x_1 = \frac{b_1}{l_{1,1}},

x_2 = \frac{b_2 - l_{2,1} x_1}{l_{2,2}},

\vdots

x_m = \frac{b_m - \sum_{i=1}^{m-1} l_{m,i}x_i}{l_{m,m}}.

A matrix equation with an upper triangular matrix U can be solved in an analogous way.