triangular matrix

1 Triangular Matrix

Let n be a positive integer.

An upper triangular matrixMathworldPlanetmath is of the form:


An upper triangular matrix is sometimes also called right triangular.

A lower triangular matrix is of the form:


A lower triangular matrix is sometimes also called left triangular.

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

A triangular matrix is a matrix that is an upper triangular matrix or lower triangular matrix. Note that some matrices, such as the identity matrixMathworldPlanetmath, are both upper and lower triangular. A matrix is upper and lower triangular simultaneously if and only if it is a diagonal matrixMathworldPlanetmath.

Triangular matrices allow numerous algorithmic shortcuts in many situations. For example, if A is an n×n triangular matrix, the equation Ax=b can be solved for x in at most n2 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 inversePlanetmathPlanetmath of a triangular matrix is a triangular matrix.

  • The product of two triangular matrices is a triangular matrix.

  • The determinantMathworldPlanetmath of a triangular matrix is the product of the diagonal elements.

  • The eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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