Triangular matrix

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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 easier 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.

1 Description
2 Special forms
3 Special properties
4 Generalizations
5 Examples
6 Application
7 See also

A matrix of the form

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

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.

The standard operations on triangular matrices conveniently preserve the triangular form: the sum and product of two upper triangular matrices is again upper triangular. The inverse of an upper triangular matrix is also upper triangular, and of course we can multiply an upper triangular matrix by a constant and it will still be upper triangular. This means that the upper triangular matrices form a subalgebra of the ring of square matrices for any given size. The analogous result holds for lower triangular matrices. Note, however, that the product of a lower triangular with an upper triangular matrix does not preserve triangularity.

A triangular matrix with zero entries on the main diagonal is strictly upper or lower triangular. All strictly triangular matrices are nilpotent.

If the entries on the main diagonal are 1, the matrix is termed unit upper/lower or normed upper/lower triangular. (However, a normed matrix is not the same as a normal matrix, and a unit matrix is not the same as the unit matrix.)

A Gauss matrix is a special form of a normed triangular matrix, where all of the off-diagonal entries are zero, except for the entries in one column. Such a matrix is also called atomic upper/lower triangular or Gauss (transformation) matrix. So an atomic lower triangular matrix is of the form

$\mathbf{L}_{i} = \begin{bmatrix} 1 & & & & & & & 0 \\ 0 & \ddots & & & & & & \\ 0 & \ddots & 1 & & & & & \\ 0 & \ddots & 0 & 1 & & & & \\ & & 0 & l_{i+1,i} & 1 & & & \\ \vdots & & 0 & l_{i+2,i} & 0 & \ddots & & \\ & & \vdots & \vdots & \vdots & \ddots & 1 & \\ 0 & \dots & 0 & l_{n,i} & 0 & \dots & 0 & 1 \\ \end{bmatrix}.$

The inverse of an atomic triangular matrix is again atomic triangular. Indeed, we have

$\mathbf{L}_{i}^{-1} = \begin{bmatrix} 1 & & & & & & & 0 \\ 0 & \ddots & & & & & & \\ 0 & \ddots & 1 & & & & & \\ 0 & \ddots & 0 & 1 & & & & \\ & & 0 & -l_{i+1,i} & 1 & & & \\ \vdots & & 0 & -l_{i+2,i} & 0 & \ddots & & \\ & & \vdots & \vdots & \vdots & \ddots & 1 & \\ 0 & \dots & 0 & -l_{n,i} & 0 & \dots & 0 & 1 \\ \end{bmatrix},$

i.e. the off-diagonal entries are replaced by their opposites.

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.

A matrix which is simultaneously triangular and normal, is also diagonal. This can be seen by looking at the diagonal entries of A^*A and AA^*, where A is a normal, triangular matrix.

The transpose of an upper triangular matrix is a lower triangular matrix and vice versa. The determinant of a triangular matrix equals the product of the diagonal entries, and the eigenvalues of a triangular matrix are 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.

Generally, operations can be performed on triangular matrices within half of the time that is needed for the same operation on a general matrix.

Because the product of two upper triangular matrices is again upper triangular, the set of upper triangular matrices forms an algebra. Algebras of upper triangular matrices have a natural generalization in functional analysis which yields nest algebras on Hilbert spaces.

The set of invertible triangular matrices form a group, and is a subgroup of all invertible matrices. The set of 2 by 2 triangular matrices is called the parabolic subgroup; 3 by 3 and larger normed triangular matrices form the Heisenberg group. Both are examples of a Borel subgroup.

Upper triangular matrices stabilize the standard flag, so the abstract (without a basis) analog of upper triangular matrices are the stabilizer of a complete flag, or alternatively a Borel subgroup.

The abstract analog of block upper triangular matrices are the stabilizer of a partial flag, or alternatively a parabolic subgroup.

Lower triangular matrices stabilize the flag generated by $e_n,\dots,e_1$ (the standard basis in reverse order). More generally, in a basis adapted to a complete flag, lower triangular matrices are the stabilizer of the orthogonal flag (given by $W_i = V_{n-i}^\perp$ ).

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

$\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 2 & 0 & 1 \\ \end{bmatrix}$

is atomic lower triangular. Its inverse is

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

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

$\begin{matrix} l_{1,1} x_1 & & & & & = & b_1 \\ l_{2,1} x_1 & + & l_{2,2} x_2 & & & = & b_2 \\ \vdots & & \vdots & \ddots & & & \vdots \\ l_{m,1} x_1 & + & l_{m,2} x_2 & + \ldots + & l_{m,m} x_m & = & b_m \\ \end{matrix}$