= Discrete orthogonality of Gaussian quadrature
{title2=$A^TWA=I,\quad AA^T=W^{-1}$}
Let $\phi_0,\ldots,\phi_n$ be <orthonormal> <polynomials> for a positive weight, and let $x_i,w_i$ define its $(n+1)$-point <Gaussian quadrature>. Put $A_{ij}=\phi_{j-1}(x_{i-1})$ and $W=\operatorname{diag}(w_0,\ldots,w_n)$. Exactness through <polynomial degree> $2n+1$ gives $A^TWA=I$. Positivity of the <quadrature weights> makes $W^{1/2}A$ a square <orthogonal matrix>, so $AA^T=W^{-1}$ also. Thus the unweighted row <inner products> of the evaluation <matrix> are $\delta_{ij}/w_{i-1}$.
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