Is there a standard notation for the set of lower triangular matrices, and if so what is it. Additionally, is there a notation for the set of strictly lower triangular matrices, and if so what is it? Thanks.
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If I were a textbook writer, I'll "draw" a lower triangle(with right angle in the lower left), like an alphabet. – Eric Sep 30 '16 at 14:56
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1using $L$ is fine, but the visual symbol is even better.. :P – Eric Sep 30 '16 at 14:57
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isn't just the column number lower than the row number? – Ben Aug 09 '19 at 20:20
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Very often the vector space of $m\times m$ lower triangular matrices with coefficients in a field $K$, or a ring $R$, is denoted by $\mathfrak{b}_m(K)$, and the subspace of strictly lower-triangualr ones by $\mathfrak{n}_m(K)$. Here the $\mathfrak{b}$ stands for the Lie algebra of the Borel subgroup $B$ of $GL_m(K)$, of lower-triangular invertible matrices, and the $\mathfrak{n}$ for its nilradical $N$. In this sense, $\mathfrak{b}_m(K)$ and $\mathfrak{n}_m(K)$ are Lie subalgebras of $\mathfrak{gl}_m(K)$, where $\mathfrak{b}_m(K)$ is solvable, and $$\mathfrak{n}_m(K)=[\mathfrak{b}_m(K),\mathfrak{b}_m(K)]$$ is nilpotent. The commutator of two matrices here is given by $[A,B]:=AB-BA$. So the notation $B$ comes from "Borel", and $N$ from Nilradical, or nilpotent.
Without the context of Lie algebras, see here.
Dietrich Burde
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