I have the expression $UD(D^2 + \lambda I)^{-1}DU^Ty$ where $U \in \mathbb{R}^{n \times p}, D \in \mathbb{R}^{p \times p}, y \in \mathbb{R}^{n \times 1}$.
Apparently it is equivalent to $$ \sum_{j=1}^p u_j \frac{d_j^2}{d_j^2 + \lambda}u_j^Ty $$ where $u_j$ is the j-th column of $U$. Is this supposed to be obvious, or are there some middle steps to get the summation?
Edit: Realized I didn't provide sufficient info. $D$ is a diagonal matrix. $U$ is a matrix with orthonormal columns.