  We use tools from $q$-calculus to study $LDL^T$ decomposition of the
Vandermonde matrix $V_q$ with coefficients $v_{i,j}=q^{ij}$. We prove that the
matrix $L$ is given as a product of diagonal matrices and the lower triangular
Toeplitz matrix $T_q$ with coefficients $t_{i,j}=1/(q;q)_{i-j}$, where
$(z;q)_k$ is the q-Pochhammer symbol. We investigate some properties of the
matrix $T_q$, in particular, we compute explicitly the inverse of this matrix.
