Prove that $Q^{T}TQ$ is symmetric and tridiagonal, where $Q,R$ is $QR$ decomposition of symmetric tridiagonal matrix $T$

Question

Let $T$ be quadratic, invertible, symmetrical tridiagonal matrix and $Q$, $R$ matrices be $QR$ decomposition of $T$. Prove that $Q^{T}TQ$ is also symmetrical and tridiagonal.

The symmetric part is easy: $(Q^{T}TQ)^{T} = Q^{T}T^{T}Q = Q^{T}TQ$. I have no clue about how to prove it is tridiagonal.

Answer 1

$Q=TR^{-1}$ can only have one non-zero lower sub-diagonal, thus the same holds for $Q^TTQ=RQ$. Add symmetry.