Roughness-matrix formula for a natural cubic smoothing spline ($A_{\lambda }=(I+\lambda K{)}^{-1}$)

For distinct ordered observations $x_i$, let $s_u$ be their natural cubic spline interpolant with values $u_i$. Its roughness is a quadratic form, $\int (s_u^{\prime \prime }{)}^2=u^{T}Ku$, where $K$ is positive semidefinite and its nullspace consists of vectors sampled from affine functions. Minimizing penalized least squares $\Vert y-u{\Vert }^2+\lambda u^{T}Ku$ gives $\hat{u}=(I+\lambda K{)}^{-1}y$. Thus the smoothing matrix is $A_{\lambda }=(I+\lambda K{)}^{-1}$ and its effective degrees of freedom are $\mathrm{tr}(A_{\lambda })$. As $\lambda \downarrow 0$ the fit approaches interpolation and the trace approaches $n$; as $\lambda \to \infty$ it approaches the least-squares affine line and the trace approaches $2$.