### 5.4 Smoothing Splines

#### Derivation of Equation (5.12)

Equal the derivative of **Equation (5.11)** as zero, we get

Put the terms related to $\theta$ on one side and the others on the other side, we get

Multiply the inverse of $N^TN+\lambda\Omega_N$ on both sides completes the derivation of **Equation (5.12)**

#### Explanations on Equation (5.17) and (5.18)

It’s a little confusing to get **Equation (5.18)** directly from **Equation (5.17)** and its original form **Equation (5.11)**. In order to give a clear explanation, here we give the proof of the equation,

which are the different terms between **Equation (5.11)** and **Equation (5.18)**.

We know that

following **Equation (5.14)** in the book.

From **Equation (5.17)**, we can get

Plug the above two equation into the right side of the equation remains to be proved, we get

which completes the proof.