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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 & & \
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} =
\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 & dots \
& & & \ddots & u_{n-1,n}\
0 & & & & u_{n,n}
\end{bmatrix}


is called upper triangular matrix or '''right triangular matrix'''.

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

  • ''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 Eigenvalue s 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

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 generalization in Functional Analysis which yields Nest Algebra s on Hilbert Space s.

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 .


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}

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

:
\begin{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
: x_1 = rac{b_1}{l_{1,1}},
: x_2 = rac{b_2 - l_{2,1} x_1}{l_{2,2}},
:: dots
: x_m = rac{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.


SEE ALSO