Let $T$ be a real symmetric Toeplitz matrix of dimension $n$. We write $T_i$ for the matrix with only the first $i$ rows and columns of $T$.
In my implementation of the Levinson algorithm I'm building a family of backward vectors $(x_i)_{1 \leq i \leq n}$ such that $$ x_1 = \left( \frac{1}{T_{11}} \right) $$ and for $2 \leq i \leq n$ $$ x_i = \lambda_i \begin{pmatrix} x_{i-1, i-1} \\ \vdots \\ x_{i-1, 1} \\ 0 \end{pmatrix} + \mu_i \begin{pmatrix} 0 \\ x_{i-1, 1} \\ \vdots \\ x_{i-1, i-1} \end{pmatrix} $$ with $\lambda_i, \mu_i \in \mathbb{R}$. We note that what Wikipedia calls forward vector is, here, the reverse of the backward one. Also they have the property that $$ T_i x_i = \begin{pmatrix} 0 \\ \vdots \\ 0 \\ 1 \end{pmatrix} $$
I want to show that the last coordinate of $x_i$ satisfy $x_{i, i} > 0$. We easily see that $$ x_{i,i} = \frac{1}{T_{11}} \prod_{k=2}^i \mu_k. $$ I think I need to do an induction on the last equation but without directly knowing the $\mu_i$ it doesn't seem easy. So I don't know where to start.