---- > [!definition] Definition. ([[strictly triangular matrix]]) > A **strictly triangular** [[matrix]] is a [[lower-triangular matrix|triangular matrix]] which has zeros on the [[diagonal]]. > The set of $n \times n$ strictly upper triangular matrices forms a [[Lie subalgebra]] of the [[general linear Lie algebra|general linear]] [[Lie algebra]] $\mathfrak{gl}_{n}$, denoted $\mathfrak{n}_{n}$.[^1] ^4365bc [^1]: $\mathfrak{n}$ stands for nilpotent. > [!basicexample] > - [[directed acyclic network has upper triangular adjacency matrix wrt vertical ordering|Every acyclic network has a strictly upper triangular adjacency matrix with respect to some indexing of its nodes]] > [!justification] It is clear that $\mathfrak{b}_{n}$ is a [[linear subspace]] of $\mathfrak{gl}_{n}$ (0 is strictly triangular, as is a sum or scaling of strictly triangulars). To confirm stability wrt the [[commutator]], it suffices to recall that the product of strictly upper triangular matrices is strictly 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 > ```