[proofplan] We diagonalize the ridge normal matrix $X^\top X + \rho I_p$ using the right singular vector basis of $X$. In that basis, inversion is scalar division by $d_j^2+\rho$ on the row-space directions and by $\rho$ on the null-space directions. Since $X^\top y$ has no component in the null space of $X$, only the first $r$ singular directions contribute. Multiplying the resulting coefficient expansion by $X$ gives the fitted-value formula with shrinkage factors $d_j^2/(d_j^2+\rho)$. [/proofplan]

[step:Diagonalize the ridge normal matrix in the right singular vector basis] Since $X = UDV^\top$, orthogonality of $U$ gives \begin{align*} X^\top X &= (UDV^\top)^\top(UDV^\top) \\ &= VD^\top U^\top UDV^\top \\ &= VD^\top DV^\top. \end{align*} Define the diagonal matrix \begin{align*} A := D^\top D + \rho I_p \in \mathbb{R}^{p \times p}. \end{align*} Then \begin{align*} X^\top X + \rho I_p = VAV^\top. \end{align*} The diagonal entries of $A$ are $d_j^2+\rho$ for $1 \le j \le r$ and $\rho$ for $r < j \le p$. Because $\rho > 0$, all diagonal entries of $A$ are positive, so $A$ is invertible. Since $V$ is orthogonal, \begin{align*} (X^\top X + \rho I_p)^{-1} = VA^{-1}V^\top. \end{align*} [/step]

[step:Expand $X^\top y$ in the right singular vector basis] For each $1 \le j \le r$, the diagonal structure of $D$ gives $D^\top u_j = d_j e_j$, where $e_j \in \mathbb{R}^p$ is the $j$-th standard basis vector. For $r < j \le n$, the corresponding singular value is zero, so no positive singular direction contributes. Hence \begin{align*} X^\top y &= VD^\top U^\top y \\ &= V\left(\sum_{j=1}^r d_j(u_j^\top y)e_j\right) \\ &= \sum_{j=1}^r d_j(u_j^\top y)v_j. \end{align*} [/step]

[step:Apply the inverse diagonal multiplier to obtain the coefficient formula] Using the previous two steps, \begin{align*} \hat{\beta}^{\mathrm{ridge}}(\rho) &= (X^\top X + \rho I_p)^{-1}X^\top y \\ &= VA^{-1}V^\top \left(\sum_{j=1}^r d_j(u_j^\top y)v_j\right). \end{align*} Since $V^\top v_j = e_j$ and $A^{-1}e_j = (d_j^2+\rho)^{-1}e_j$ for $1 \le j \le r$, we get \begin{align*} \hat{\beta}^{\mathrm{ridge}}(\rho) &= \sum_{j=1}^r d_j(u_j^\top y)V A^{-1}e_j \\ &= \sum_{j=1}^r \frac{d_j}{d_j^2+\rho}(u_j^\top y)Ve_j \\ &= \sum_{j=1}^r \frac{d_j}{d_j^2+\rho}(u_j^\top y)v_j. \end{align*} [/step]

[step:Multiply by $X$ to obtain the fitted-value formula] For $1 \le j \le r$, the [singular value decomposition](/theorems/3071) gives \begin{align*} Xv_j = UDV^\top v_j = UDe_j = d_j u_j. \end{align*} Therefore \begin{align*} X\hat{\beta}^{\mathrm{ridge}}(\rho) &= X\left(\sum_{j=1}^r \frac{d_j}{d_j^2+\rho}(u_j^\top y)v_j\right) \\ &= \sum_{j=1}^r \frac{d_j}{d_j^2+\rho}(u_j^\top y)Xv_j \\ &= \sum_{j=1}^r \frac{d_j^2}{d_j^2+\rho}(u_j^\top y)u_j. \end{align*} This is the stated fitted-value expansion, and the proof is complete. [/step]