Triangular Matrix

Let be a positive integer.

An upper triangular matrix is of the form:

$\displaystyle \begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} \ 0 ... ...vdots & \vdots & \ddots & \vdots \ 0 & 0 & 0 & \cdots & a_{nn} \end{bmatrix}$

An upper triangular matrix is sometimes also called right triangular.

A lower triangular matrix is of the form:

$\displaystyle \begin{bmatrix} a_{11} & 0 & 0 & \cdots & 0 \ a_{21} & a_{22} &... ...& \ddots & \vdots \ a_{n1} & a_{n2} & a_{n3} & \cdots & a_{nn} \end{bmatrix}$

A lower triangular matrix is sometimes also called left triangular.

Note that upper triangular matrices and lower triangular matrices must be square matrices.

A 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 is an $n\times n$ triangular matrix, the equation can be solved for in at most 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.

Properties

Triangular matrices have the following properties (prefix “triangular” with either “upper” or “lower” uniformly):

The inverse of a triangular matrix is a triangular matrix.
The product of two triangular matrices is a triangular matrix.
The determinant of a triangular matrix is the product of the diagonal elements.
The eigenvalues of a triangular matrix are the diagonal elements.

The last two properties follow easily from the cofactor expansion of the triangular matrix.

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Also defines:

upper triangular, lower triangular, upper triangular matrix, lower triangular matrix, right triangular, right triangular matrix, left triangular, left triangular matrix

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Attachments:: theorem for normal triangular matrices (Theorem) by Mathprof

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Cross-references: cofactor expansion, eigenvalues, diagonal, determinant, product, inverse, properties, LU-factorization, Cholesky factorization, matrix factorization, linear algebra, operations, equation, diagonal matrix, identity matrix, matrix, square matrices, integer, positive
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This is version 9 of triangular matrix, born on 2002-01-16, modified 2007-04-19.
Object id is 1483, canonical name is TriangularMatrix.
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Classification:

AMS MSC:	15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
	65-00 (Numerical analysis :: General reference works )

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