My course notes on iterative methods make the following claim
If $T$ is a symmetric, positive definite and tridiagonal $k \times k$ matrix, it has a $LU$ factorisation without pivoting. We can write $T = LU$ with $$ L = \begin{bmatrix} \delta_0 \\ \phi_0 & \delta_1 \\ & \phi_1 & \ddots \\ & & \ddots & \ddots \\ & & & \phi_{k - 2} & \delta_{k-1} \\ \end{bmatrix} \quad U = \begin{bmatrix} 1 & \epsilon_{0} \\ & 1 & \epsilon_1 \\ & & \ddots & \ddots \\ & & & \ddots & \epsilon_{k-2} \\ & & & & 1 \\ \end{bmatrix} $$ for certain real numbers $\delta_i$, $\phi_i$, $\epsilon_i$.
No proof or reference is given whatsoever. I found these notes which give an algorithm to determine the numbers $\delta_i$, $\phi_i$ and $\epsilon_i$ assuming that they exist, but clearly this is not the case for any symmetric, tridiagonal matrix. How does the positive definiteness ensure that such a factorisation exists? It's just a theoretical result that's needed, I don't really need an algorithm.