triangular matrix
TriangularMatrix
2002-01-16 07:46:11
2007-04-19 12:22:47
Definition
matrix
upper triangular
lower triangular
upper triangular matrix
lower triangular matrix
right triangular
right triangular matrix
left triangular
left triangular matrix
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\section{Triangular Matrix}
Let $n$ be a positive integer.
An \emph{upper triangular matrix} is of the form:
$$ \begin{bmatrix}
a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \\
0 & a_{22} & a_{23} & \cdots & a_{2n} \\
0 & 0 & a_{33} & \cdots & a_{3n} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & a_{nn}
\end{bmatrix} $$
An upper triangular matrix is sometimes also called \emph{right triangular}.
A \emph{lower triangular matrix} is of the form:
$$ \begin{bmatrix}
a_{11} & 0 & 0 & \cdots & 0 \\
a_{21} & a_{22} & 0 & \cdots & 0 \\
a_{31} & a_{32} & a_{33} & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn}
\end{bmatrix} $$
A lower triangular matrix is sometimes also called \emph{left triangular}.
Note that upper triangular matrices and lower triangular matrices must be square matrices.
A \emph{triangular matrix} is a matrix that is an upper triangular matrix or lower triangular matrix. Note that some matrices, such as the identity matrix, are both upper and lower triangular. A matrix is upper and lower triangular simultaneously if and only if it is a diagonal matrix.
Triangular matrices allow numerous algorithmic shortcuts in many situations. For example, if $A$ is an $n\times n$ triangular matrix, the equation $Ax=b$ can be solved for $x$ in at most $n^2$ operations.
In fact, triangular matrices are so useful that much computational linear algebra begins with factoring (or decomposing) a general matrix or matrices into triangular form. Some matrix factorization methods are the Cholesky factorization and the LU-factorization. Even including the factorization step, enough later operations are typically avoided to yield an overall time savings.
\section{Properties}
Triangular matrices have the following properties (\PMlinkescapetext{prefix} ``triangular'' with either ``upper'' or ``lower'' uniformly):
\begin{itemize}
\item The inverse of a triangular matrix is a triangular matrix.
\item The product of two triangular matrices is a triangular matrix.
\item The determinant of a triangular matrix is the product of the diagonal elements.
\item The eigenvalues of a triangular matrix are the diagonal elements.
\end{itemize}
The last two properties follow easily from the cofactor expansion of the triangular matrix.