Matrix Decomposition?

Question

How do you prove that $$ GL(n,\mathbb{C})/ B(n, \mathbb{C}) \sim U(n)/T(n) $$. Where $ B(n,\mathbb{C}) $ is the group of invertible upper triangular matrices, and $ T(n) $ is the group of diagonal unitary matrices? One way is to show that these are both the flag manifold over $ \mathbb{C}$. But is there a straightforward matrix way that I don't know?

Answer 1

This follows from Gram-Schmidt decomposition. More precisely, choose an inner product and an orthonormal basis $ \{ e_1, \cdots, e_n \} $ on $ \mathbb{C}^n $. let $ M \in GL(n,\mathbb{C}) $ and let $ \{ e_1', e_2', \cdots, e_n' \} $ be Gram Schmidt applied to the sequence $ M e_1, M e_2, \cdots $. Note that by Gram-Schmidt, $ M e_n $ is in the span of $ \langle e_n', e_{n-1}', e_{n-2}',\cdots \rangle $. This implies that the transformation $ e_i' \rightarrow M e_i $ is precisely upper triangular. Finally the transformation $ e_i \rightarrow e_i $ is exactly an element of $ U(n)/T(n) $.

Sorry for posting and answering my own question. I was not thinking clearly yesterday.