---- > [!definition] Definition. ([[upper-triangular matrix]]) > A [[matrix]] is called **upper-triangular** if all the entries below the [[diagonal]] are $0$. $\begin{bmatrix}* & * & * & * \\0 & * & * & * \\0 & 0 & * & * \\0 & 0 & 0 & * \end{bmatrix}$ The set of all $n \times n$ upper-triangular matrices forms a [[Lie subalgebra]] of the [[general linear Lie algebra|general linear]] [[Lie algebra]] $\mathfrak{gl}_{n}$, denoted $\mathfrak{b}_{n}$. ^definition > [!equivalence] > [[characterization of upper triangular matrices]] ^equivalence > [!justification] Justifying that $\mathfrak{b}_{n}$ is a [[Lie subalgebra]] of $\mathfrak{gl}_{n}$. It is clear that $\mathfrak{b}_{n}$ is a [[linear subspace]] of $\mathfrak{gl}_{n}$ (0 is triangular, as is a sum or scaling of triangulars). To confirm stability wrt the [[commutator]], it suffices to recall that the product of upper triangular matrices is upper triangular. ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```