Editing Triangular matrix (section)

== Algebras of triangular matrices ==
[[File:Cyclic group Z4; Cayley table; powers of Gray code permutation (small).svg|thumb|[[Logical matrix|Binary]] lower unitriangular [[Toeplitz matrix|Toeplitz]] matrices, multiplied using [[Finite field|'''F'''<sub>2</sub>]] operations. They form the [[Cayley table]] of [[cyclic group|Z<sub>4</sub>]] and correspond to [[v:Gray code permutation powers#4 bit|powers of the 4-bit Gray code permutation]].]]

Upper triangularity is preserved by many operations: 
* The sum of two upper triangular matrices is upper triangular.
* The product of two upper triangular matrices is upper triangular.
* The [[inverse matrix|inverse]] of an upper triangular matrix, if it exists, is upper triangular.
* The product of an upper triangular matrix and a scalar is upper triangular. 

Together these facts mean that the upper triangular matrices form a [[subalgebra]] of the [[associative algebra]] of square matrices for a given size. Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the [[Lie algebra]] of square matrices of a fixed size, where the [[Lie bracket]] [''a'', ''b''] given by the [[commutator#Ring theory|commutator]] {{nowrap|''ab − ba''}}. The Lie algebra of all upper triangular matrices is a [[solvable Lie algebra]]. It is often referred to as a [[Borel subalgebra]] of the Lie algebra of all square matrices.

All these results hold if ''upper triangular'' is replaced by ''lower triangular'' throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangular matrix can be any matrix; the product of a lower triangular with an upper triangular matrix is not necessarily triangular either.

The set of unitriangular matrices forms a [[Lie group]].

The set of strictly upper (or lower) triangular matrices forms a [[nilpotent Lie algebra]], denoted <math>\mathfrak{n}.</math> This algebra is the [[derived Lie algebra]] of <math>\mathfrak{b}</math>, the Lie algebra of all upper triangular matrices; in symbols, <math>\mathfrak{n} = [\mathfrak{b},\mathfrak{b}].</math> In addition, <math>\mathfrak{n}</math> is the Lie algebra of the Lie group of unitriangular matrices.

In fact, by [[Engel's theorem]], any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable.

Algebras of upper triangular matrices have a natural generalization in [[functional analysis]] which yields [[nest algebra]]s on [[Hilbert space]]s.

{{see also|Affine group}}

===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.

=== Examples ===
The group of 2×2 upper unitriangular matrices is [[isomorphic]] to the [[Abelian group|additive group]] of the field of scalars; in the case of complex numbers it corresponds to a group formed of parabolic [[Möbius transformation]]s; the 3×3 upper unitriangular matrices form the [[Heisenberg group]].