# Natural Splines and Smoother Matrix

In the context of smoothing splines, one can show that the Reinsch form is given by:

$$\hat{y} = N (N^{T}N +\lambda \Omega)^{-1}N^{T} y = (I+ \lambda K)^{-1}y$$

where (1) $$K = (N^{T})^{-1}\Omega N^{-1}$$

(for details regarding notation, see Invertibility in Reinsch form Derivation (Smoothing Splines))

In the textbook by Green and Silverman (Nonparametric Regression and Generalized Linear Models: A roughness penalty approach: pp. 12/13), I find the following expression for $$K$$:

(2) $$K = QR^{-1}Q^{T}$$

where $$Q$$ and $$R$$ are tridiagonal matrices.

Let $$h_i = t_{i+1}-t_i$$ for $$i = 1, \dots, n-1$$.($$t_{i}$$ are the knot points)

$$Q$$ is the $$n x (n-2)$$ matrix with entries $$q_{j-1,j} = h_{j-1}^{-1}$$, $$q_{j,j} = -h_{j-1}^{-1}-h_{j}^{-1}$$ and $$q_{j+1,j} = h_{j+1}^{-1}$$ for $$j= 2, \dots, n-1$$ and $$q_{ij} = 0$$ for $$|i-j|\geq2$$

$$R$$ is a $$(n-2) x (n-2)$$ matrix consisting of $$r_{ii} = 1/3(h_{i-1}+h_{i})$$ for $$i=,2,\dots, n-1$$ and $$r_{i, i+1} = r{i+1,i} = 1/6 h_{i}$$ for $$i = 2, \dots, n-2$$ and $$r_{ij} = 0$$ for $$|i-j| \geq 2$$

What is the difference between (1) and (2)? How can one show that those expressions are equivalent or are they are already equivalent?