$**$Question $2$: Show that the efficient expressions for the effective degrees of freedom $($EDF$)$ and the parameter $\backslash \backslash (\backslash \backslash$beta $\backslash \backslash )$ in the context of thin plate splines yield the same results as the original, less efficient expressions.$**$

Thin plate splines are a non$-$parametric method used for estimating functions from data. The model involves an unknown smooth function $\backslash \backslash ($ f $\backslash \backslash )$ representing the relationship between observed variables $\backslash \backslash ($ x$\backslash \_$i $\backslash \backslash )$ and $\backslash \backslash ($ y$\backslash \_$i $\backslash \backslash ),$ with an error term $\backslash \backslash (\backslash \backslash$epsilon$\backslash \_$i $\backslash \backslash ).$

The model is usually estimated by minimizing a weighted sum of lack of fit and wiggliness of $\backslash \backslash ($ f $\backslash \backslash ).$ To achieve this, a bivariate smoothing is applied using a thin plate spline function. The efficient expressions for the effective degrees of freedom $($EDF$)$ and the parameter $\backslash \backslash (\backslash \backslash$beta $\backslash \backslash )$ are derived using a QR decomposition and an eigendecomposition.

$**$Original Expressions:$**$

$1\backslash .$ Original EDF Expression:

$\backslash \backslash \backslash [$ EDF $=\backslash \backslash$text$\{$trace$\}\backslash \backslash$left$(($X$^$T X $+\backslash \backslash$lambda S$)^\{-1\}$ X$^$T X$\backslash \backslash$right$)\backslash \backslash \backslash ]$

$2\backslash .$ Original $\backslash \backslash (\backslash \backslash$beta $\backslash \backslash )$ Expression:

$\backslash \backslash \backslash [\backslash \backslash$beta $=($X$^$T X $+\backslash \backslash$lambda S$)^\{-1\}$ X$^$T y $\backslash \backslash \backslash ]$

$**$Efficient Expressions:$**$

$1\backslash .$ Efficient EDF Expression:

$\backslash \backslash \backslash [$ EDF $=\backslash \backslash$text$\{$trace$\}\backslash \backslash$left$(($I $+\backslash \backslash$lambda $\backslash \backslash$Lambda$)^\{-1\}\backslash \backslash$right$)\backslash \backslash \backslash ]$

$2\backslash .$ Efficient $\backslash \backslash (\backslash \backslash$beta $\backslash \backslash )$ Expression:

$\backslash \backslash \backslash [\backslash \backslash$beta $=$ R$^\{-1\}$ U $($I $+\backslash \backslash$lambda $\backslash \backslash$Lambda$)^\{-1\}$ U$^$T Q$^$T y $\backslash \backslash \backslash ]$

$**$Additional Information:$**$

$\backslash -\backslash \backslash ($ U $\backslash \backslash )$ is obtained from the eigendecomposition $\backslash \backslash ($ U $\backslash \backslash$Lambda U$^$T $=$ R$^\{-$T$\}$ S R$^\{-1\}\backslash \backslash ).$

$\backslash -\backslash \backslash ($ S $\backslash \backslash )$ is a diagonal matrix with eigenvalues.

$\backslash -\backslash \backslash ($ R $\backslash \backslash )$ is the matrix obtained from the QR decomposition of $\backslash \backslash ($ X $\backslash \backslash ).$

$\backslash -$ The matrix $\backslash \backslash ($ U$^$T R$^\{-$T$\}$ R$^\{-1\}$ U $\backslash \backslash )$ is crucial in the efficient expressions.

The objective is to demonstrate step by step, using mathematical details, that these two sets of expressions for EDF and $\backslash \backslash (\backslash \backslash$beta $\backslash \backslash )$ are equivalent. This involves manipulating matrix expressions, applying properties of inverses and traces, and leveraging the QR decomposition and eigendecomposition.

Please provide a detailed proof, explaining the rules and properties used at each step, to show the equivalence between the original and efficient expressions for EDF and $\backslash \backslash (\backslash \backslash$beta $\backslash \backslash ).$