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''.

Contents

Descripion

Special forms

Special properties

Generalizations

Examples

Application

See also

Descripion

A matrix of the form
:$mathbf\{L\}=$
egin{bmatrix}
l_{1,1} & & & & 0 \
l_{2,1} & l_{2,2} & & & \
l_{3,1} & l_{3,2} & ddots & & \
dots & dots & 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
:$mathbf\{U\}\; =$
egin{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 & dots \
& & & ddots & u_{n-1,n}\
0 & & & & u_{n,n}
end{bmatrix}

is called 'upper triangular matrix' or 'right triangular matrix'.
Certain operations on triangular matricies conveniently preserve the triangular form. For example: The product of two upper triangular matricies is upper triangular. The inverse of an upper triangular matrix is upper triangular. Likewise for matricies that are all lower triangular. However note that the product of a ''lower'' triangular with an ''upper'' triangular matrix does ''not'' preserve triangularity.

Special forms

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
:$mathbf\{L\}\_\{i\}\; =$
egin{bmatrix}
1 & & & & & & & 0 \
0 & ddots & & & & & & \
0 & ddots & 1 & & & & & \
0 & ddots & 0 & 1 & & & & \
& & 0 & l_{i+1,i} & 1 & & & \
dots & & 0 & l_{i+2,i} & 0 & ddots & & \
& & dots & dots & dots & 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
:$mathbf\{L\}\_\{i\}^\{-1\}\; =$
egin{bmatrix}
1 & & & & & & & 0 \
0 & ddots & & & & & & \
0 & ddots & 1 & & & & & \
0 & ddots & 0 & 1 & & & & \
& & 0 & -l_{i+1,i} & 1 & & & \
dots & & 0 & -l_{i+2,i} & 0 & ddots & & \
& & dots & dots & dots & 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.

Special properties

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.

Generalizations

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 $W\_i\; =\; V\_\{n-i\}^perp$).

Examples

The matrix
:$$
egin{bmatrix}
1 & 4 & 2 \
0 & 3 & 4 \
0 & 0 & 1 \
end{bmatrix}

is upper triangular and
:$$
egin{bmatrix}
1 & 0 & 0 \
2 & 8 & 0 \
4 & 9 & 7 \
end{bmatrix}

is lower triangular.
The matrix
:$$
egin{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
:$$
egin{bmatrix}
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & -4 & 1 & 0 \
0 & -2 & 0 & 1 \
end{bmatrix}.

Application

A matrix equation in the form
:$mathbf\{L\}mathbf\{x\}\; =\; mathbf\{b\}$
or
:$mathbf\{U\}\; mathbf\{x\}\; =\; mathbf\{b\}$
is very easy to solve. The matrix equation ''Lx'' = ''b'' can be written as a system of linear equations
:$$
egin{matrix}
l_{1,1} x_1 & & & & & = & b_1 \
l_{2,1} x_1 & + & l_{2,2} x_2 & & & = & b_2 \
dots & & dots & ddots & & & dots \
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
:
:
::
:
A matrix equation with an upper triangular matrix ''U'' can be solved in an analogous way. This process is called 'back substitution'.

See also

★ Gaussian elimination

★ QR decomposition

★ Hessenberg matrix

★ Tridiagonal matrix

★ Invariant subspace