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?

Thanks in advance!


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