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Importing Wikidata short description: "Special kind of square matrix" (Shortdesc helper) |
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===Borel subgroups and Borel subalgebras===
{{main|Borel subgroup|Borel subalgebra}}
The set of invertible triangular matrices of a given kind (upper or lower) forms a [[group (mathematics)|group]], indeed a [[Lie group]], which is a subgroup of the [[general linear group]] of all invertible matrices. A triangular matrix is invertible precisely when its diagonal entries are invertible (non-zero).
Over the real numbers, this group is disconnected, having <math>2^n</math> components accordingly as each diagonal entry is positive or negative. The identity component is invertible triangular matrices with positive entries on the diagonal, and the group of all invertible triangular matrices is a [[semidirect product]] of this group and the group of [[Diagonal matrix|diagonal matrices]] with <math>\pm 1</math> on the diagonal, corresponding to the components.
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