 Solution

Set $W=\mathrm{diag}(w_0,\dots ,w_n)$. For column indices $j,k\in \{1,\dots ,n+1\}$, the product $P_{j-1}{P}_{k-1}$ has polynomial degree at most $2n$, so exactness of Gauss-Legendre quadrature and orthonormality giveThus $Q=W^{1/2}A$ is a square matrix with $Q^{T}Q=I$, and therefore is an orthogonal matrix: $Q{Q}^T=I$ as well. It follows thatEquivalently, the rows of the generalized Vandermonde matrix are orthogonal andThe normalization matters: here $P_{n+1}=\sqrt{(2n+3)/2}{L}_{n+1}$, where $L_{n+1}$ is the conventional Legendre polynomial used above. Thus a formula containing only the nodes and the question's orthonormal polynomials isThis identity is the discrete orthogonality of Gaussian quadrature.