triangular matrix

1 Triangular Matrix

Let $n$ be a positive integer.

An 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 right triangular.

A 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 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 $A$ 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.

2 Properties

Triangular matrices have the following properties ( “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.

Title	triangular matrix
Canonical name	TriangularMatrix
Date of creation	2013-03-22 12:11:40
Last modified on	2013-03-22 12:11:40
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	13
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 65-00
Classification	msc 15-00
Defines	upper triangular
Defines	lower triangular
Defines	upper triangular matrix
Defines	lower triangular matrix
Defines	right triangular
Defines	right triangular matrix
Defines	left triangular
Defines	left triangular matrix