Solving for $L$, where $P_n \cdots P_1 = I - XLX^T$, with $\Vert x_i \Vert_2=1$, $P_i := I - 2x_i x_i^T$, $X=[x_1 \cdots x_n]$ 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
$$
\begin{align}
P_n \cdots P_1 - I &= (I - 2x_nx_n^T) \cdots (I - 2x_1x_1^T) - I\\
&= - L_{11} x_1x_1^T - L_{21}x_2x_1^T - \cdots -L_{nn}x_nx_n^T
\end{align}
$$
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?
 A: You can try to construct the matrix $L$ recursively.
Let's we write $X_k:=[x_1,\ldots,x_k]$.
Assume tat we have already a matrix $L_k$ such that
$$
P_k\cdots P_1=I-X_kL_kX_k^T.
$$
Now let's see what happens if we multiply this by $P_{k+1}$:
$$
\begin{split}
P_{k+1}\cdots P_1
&=
(I-2x_{k+1}x_{k+1}^T)(I-X_kL_kX_k^T)\\
&=
I-2x_{k+1}x_{k+1}^T-X_kL_kX_k^T+2x_{k+1}x_{k+1}^TX_kL_kX_k^T
\end{split}
$$
We would like to express this as follows:
$$
\begin{split}
P_{k+1}\cdots P_1
&=
I-X_{k+1}L_{k+1}X_{k+1}^T\\
&=
I-[X_k,x_{k+1}]\begin{bmatrix}\tilde{L}_{k+1}&0\\l_{k+1}^T&\lambda_{k+1}\end{bmatrix}[X_k,x_{k+1}]^T\\
&=
I-X_k\tilde{L}_{k+1}X_k^T-x_{k+1}l_{k+1}^TX_k^T-\lambda_{k+1}x_{k+1}x_{k+1}^T
\end{split}
$$
Comparing terms in the two expressions, we get
$$
L_{k+1}
=\begin{bmatrix}
L_k & 0\\
-2y_k^TL_k & 2
\end{bmatrix},
$$
where $y_k:=X_k^Tx_{k+1}$. This gives a recursive formula for $L=L_n$.
If you are after an explicit formula, you can try to use this recursion and find $L$ in terms of the coefficients $y_1,\ldots,y_{n-1}$.
