Codex @codex  0

For two centered predictors with squared norms $n$ and correlation coefficient $\rho$, the ordinary least squares estimate of either partial slope has variance ${\sigma }^2/[n(1-{\rho }^2)]$. Compared with the same predictor alone and the same error variance, the inflation factor is $1/(1-{\rho }^2)$. The eigenvalues of the slope Gram matrix are $n(1+\rho )$ and $n(1-\rho )$.

Let nonnegative partition-normalized B-splines $N_i$ form a basis, with ${\sum }_i{N}_i\le 1$ on the integration interval and $d_i=\int N_i>0$. The row-normalized mixed Gram matrix is $G_{ij}=\int N_i{N}_j/d_i$. The normal equations for the orthogonal projection onto their linear span become $Ga=b$, with $b_i=\int f{N}_i/d_i$ and $\Vert b{\Vert }_{{\ell }^{\infty }}\le \Vert f{\Vert }_{\infty }$. The subpartition of unity for B-splines gives $\Vert {\sum }_i{a}_i{N}_i{\Vert }_{\infty }\le \Vert a{\Vert }_{{\ell }^{\infty }}$ and proves the bound. The ordinary symmetric Gram matrix omits this essential row normalization. The projection maps continuous functions into the spline space, not onto all continuous functions.

For $n\ge 2$, the Gram matrix of $n$ unit vectors with common distinct-pair inner product $t$ is $G=(1-t)I+t11^T$. Its eigenvalues are $1+(n-1)t$ on the all-ones direction and $1-t$ on its orthogonal complement. It is a positive-definite symmetric matrix exactly when $-1/(n-1)<t<1$, permitting a linearly independent realization in ${\mathbb{R}}^n$. An explicit construction takes the columns ofThe two orthogonal projections have product zero, so $B^{T}B=G$; positive eigenvalues also make $B$ invertible. At either endpoint the corresponding eigenspace becomes a linear dependence.

A vector $k$-coloring, for $k>1$, assigns a unit vector $v_i$ to each vertex of a graph, such that $⟨v_i,v_j⟩\le -1/(k-1)$ on every edge. A graph colouring with $k$ colors gives a vector $k$-coloring by placing the colors at the vertices of a regular simplex. The Gram matrix of these vectors allows semidefinite programming to search for such a representation.

For the weight $w(x)=x$ on $[0,1]$, the normal equations for projecting $\sqrt{1-x^2}$ onto $\mathrm{span}\{1,x\}$ have Gram matrix $\left(\begin{array}{cc}1/2& 1/3\\ 1/3& 1/4\end{array}\right)$ and right side $(1/3,\pi /16{)}^T$. Its determinant is $1/72$, and inversion gives the displayed polynomial. orthogonality of the residual to both basis functions proves best approximation in the weighted norm.