I don't understand meaning of "since a symmetric tridiagonal matrix contains only two distinct vectors" I write example for symmetric tridiagonal matrix. The 1st row vector and 1st column vector are the same, the 2nd row vector and 2nd column vector are the same. This way they are all eliminated. What does "contains only two distinct vectors" mean?
The subdiagonal elements and the superdiagonal elements can be represented as the vector $(\beta_1, \beta_2, \dots \beta_{n-1})$. It has length $n-1$. The diagonal elements can be represented as the vector $(\alpha_1, \alpha_2, \dotsc, \alpha_n)$. It has length $n$. Tridiagonal solvers such dpttrs (based on the LDLT-factorization) from LAPACK capitalize on this fact to reduce the storage requirement from $n^2$ words of memory needed for a dense matrix to a mere $2n-1$ words of memory.
Personally, I would prefer to write: "since a symmetric tridiagonal matrix can be represented using only two distinct vectors, it customary to replace the generic notation ...", but this is a matter of taste.