Triangular matrix: Difference between revisions

Revision as of 21:17, 25 June 2005

Revision as of 14:07, 30 June 2005

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A matrix which is simultaneously upper and lower triangular is [[diagonal matrix|diagonal]]. The [[identity matrix]] is the only matrix which is both normed upper and lower triangular.

The product of two upper triangular matrices is upper triangular, so the set of upper triangular matrices forms an [[associative algebra|algebra]]. Algebras of upper triangular matrices have a natural generalisation in [[functional analysis]] which yields [[nest algebra]]s.

The [[transpose]] of a upper triangular matrix is a lower triangular matrix and vice versa. The [[determinant]] of a triangular matrix equals the product of the diagonal entries.

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Generally, operations can be performed on triangular matrices within half the time.

==Generalizations==

The product of two upper triangular matrices is upper triangular, so the set of upper triangular matrices forms an [[associative algebra|algebra]]. Algebras of upper triangular matrices have a natural generalisationgeneralization in [[functional analysis]] which yields [[nest algebra]]s on [[Hilbert space]]s.

The set of invertible triangular matrices form a [[group (mathematics)|group]]. It is a subgroup of all invertible matrices, and is called the [[parabolic subgroup|parabolic]] or [[Borel subgroup]].

== Examples ==