Upper Triangular Matrix Website Links For
Matrix 		 

Information About

Upper Triangular Matrix
  CATEGORIES ABOUT TRIANGULAR MATRIX
   
numerical linear algebra
   
matrices
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



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