Discrete orthogonality of Gaussian quadrature ($A^{T}WA=I,A{A}^T=W^{-1}$)

Let ${\varphi }_0,\dots ,{\varphi }_n$ be orthonormal polynomials for a positive weight, and let $x_i,w_i$ define its $(n+1)$-point Gaussian quadrature. Put $A_{ij}={\varphi }_{j-1}(x_{i-1})$ and $W=\mathrm{diag}(w_0,\dots ,w_n)$. Exactness through polynomial degree $2n+1$ gives $A^{T}WA=I$. Positivity of the quadrature weights makes $W^{1/2}A$ a square orthogonal matrix, so $A{A}^T=W^{-1}$ also. Thus the unweighted row inner products of the evaluation matrix are ${\delta }_{ij}/w_{i-1}$.