Editing Triangular matrix (section)

===Borel subgroups and Borel subalgebras===
{{main|Borel subgroup|Borel subalgebra}}
The set of invertible triangular matrices of a given kind (lower or upper) 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.

The [[Lie algebra]] of the Lie group of invertible upper triangular matrices is the set of all upper triangular matrices, not necessarily invertible, and is a [[solvable Lie algebra]]. These are, respectively, the standard [[Borel subgroup]] ''B'' of the Lie group GL<sub>''n''</sub> and the standard [[Borel subalgebra]] <math>\mathfrak{b}</math> of the Lie algebra gl<sub>''n''</sub>.

The upper triangular matrices are precisely those that stabilize the [[Flag (linear algebra)|standard flag]]. The invertible ones among them form a subgroup of the general linear group, whose conjugate subgroups are those defined as the stabilizer of some (other) complete flag. These subgroups are [[Borel subgroup]]s. The group of invertible lower triangular matrices is such a subgroup, since it is the stabilizer of the standard flag associated to the standard basis in reverse order.

The stabilizer of a partial flag obtained by forgetting some parts of the standard flag can be described as a set of block upper triangular matrices (but its elements are ''not'' all triangular matrices). The conjugates of such a group are the subgroups defined as the stabilizer of some partial flag. These subgroups are called parabolic subgroups.