Another question brought this up. The only definition I have ever seen for a matrix being upper triangular is, written in component forms, "all the components below the main diagonal are zero." But of course that property is basis dependent. It is not preserved under change of basis.
Yet it doesn't seem as if it would be purely arbitrary because the product of upper triangular matrices is upper triangular, and so forth. It has closure. Is there some other sort of transformation besides a basis transformation that might be relevant here? It seems as if a set of matrices having this property should have some sort of invariants.
Is there some sort of isomorphism between the sets of upper triangular matrices in different bases?
Best Answer
Many true things can be said about upper-triangular matrices, obviously... :)
In my own experience, a useful more-functional (rather than notational) thing that can be said is that the subgroup of $GL_n$ consisting of upper-triangular matrices is the stabilizer of the flag (nested sequence) of subspaces consisting of the span of $e_1$, the span of $e_1$ and $e_2$, ... with standard basis vectors.
Concretely, this means the following. The matrix multiplication of a triangular matrix $A$ and $e_1$, $Ae_1$, is equal to a multiple of $e_1$, right? However, $Ae_2$ is more than a multiple of $e_2$: it can be any linear combination of $e_1$ and $e_2$. Generally, if you set $V_i= \operatorname{span}(e_1, \ldots, e_i) $, try to show that $A$ is upper triangular if and only if $A(V_i) \subseteq V_i$. The nested sequence of spaces
$$ 0 = V_0 \subset V_1 \subset \ldots \subset V_n = \mathbb{R}^n$$
is called a flag of the total space.
One proves a lemma that any maximal chain of subspaces can be mapped to that "standard" chain by an element of $GL_n$. In other words, no matter which basis you are using: being triangular is intrinsically to respect a flag with $\dim(V_i) = i$ (the last condition translate the maximality of the flag).
As Daniel Schepler aptly commented, while an ordered basis gives a maximal flag, a maximal flag does not quite specify a basis. There are more things that can be said about flags versus bases... unsurprisingly... :)