QR decomposition and Gram Schmidt process

Question

Let $A\in M_n(\mathbb{R})$, invertible matrix. I need to decompose $A=QR$ where $Q$ is an orthogonal matrix and $R$ is an upper triangular.

Now, if $A=[a_1,\ldots a_n]$ and we apply Gram Schmidt process on it's columns, we get:

$u_1 = a_1 \\ u_2 = a_2 - \frac{\langle u_1,a_2 \rangle }{\|u_1\|^2}u_1 \\ u_k = a_k - \sum_{j=1}^{k-1} \frac{\langle u_j, a_k \rangle }{\|u_j\|^2}u_j$

Rearranging the terms:

$a_1 = u_1 \\ a_2 = u_2 + \frac{\langle u_1,a_2 \rangle }{\|u_1\|^2}u_1 \\ a_k = u_k + \sum_{j=1}^{k-1} \frac{\langle u_j, a_k \rangle }{\|u_j\|^2}u_j$

Now, before moving into a matrix multipication form, I need to do something with the $\{u_1,\ldots, u_n\}$, representing them in a similar form to the other terms.

How?

Answer 1

I'm not sure I understand. The $u_i$ form the columns of the matrix $Q$. The coefficient on $u_j$ in the equation for $a_i$ is $r_{ij}$.