information:filtering, structural:reformatting
Algebras of triangular matrices

Upper triangularity is preserved by many operations:

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 given by the commutator 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 n. {\displaystyle {\mathfrak {n}}.} This algebra is the derived Lie algebra of b {\displaystyle {\mathfrak {b}}}, the Lie algebra of all upper triangular matrices; in symbols, n =. {\displaystyle {\mathfrak {n}}=.} In addition, n {\displaystyle {\mathfrak {n}}} 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 algebras on Hilbert spaces.