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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?

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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 $y_1,\ldots,y_{n-1}$.

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