Solving for $L$, where $P_n \cdots P_1 = I – XLX^T$, with $\Vert x_i \Vert_2=1$, $P_i := I – 2x_i x_i^T$


Problem

Solve for $ L$ , where $ P_n \cdots P_1 = I – XLX^T$

where $ \Vert x_i \Vert_2=1$ , $ P_i := I – 2x_i x_i^T$ , and $ X_{m \times n} = [x_1 |\cdots | x_n]$ .


Try

Note that $ L$ is a lower triangular matrix. Denoting $ (i,j)$ component of $ L$ as $ L_{ij}$ , let us find $ L_{ij}$ ‘s explicitly. We have

$ $ \begin{align} P – I &= (I – 2x_nx_n^T) \cdots (I – 2x_1x_1^T) – I\ &= – L_{11} x_1x_1^T – L_{21}x_2x_1^T – \cdots -L_{nn}x_nx_n^T \end{align} $ $

where we can note that $ -L{ij}$ is the coefficient for $ x_ix_j^T$ in $ (I – 2x_nx_n^T) \cdots (I – 2x_1x_1^T)$ .

Let us observe more carefully. For $ L_{kk}$ ‘s, we have

$ $ L_{kk} = 2 $ $

since these terms only come from $ I \cdots I (-2x_kx_k^T) I \cdots I$ . Next, we note that

$ $ L_{(k+1)k} = -2^2 $ $

and, next,

$ $ -L_{(k+2)k} = 2^2 – 2^3(v_{k+2}^Tv_{k+1})(v_{k+1}^Tv_k) $ $

but I cannot see any simpler rule for $ L_{ij}$ ‘s… Anyone to help me with finding all the $ L_{ij}$ s?