## 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?