Since \lambda \geq 0 , it means that compared to OLS, we have
\frac{\sigma_j^2}{\sigma_j^2+\lambda} \leq 1.Ridge regression finds the coordinates of \boldsymbol{y} with respect to the orthonormal basis \boldsymbol{U} , it then shrinks the coordinates by \frac{\sigma_j^2}{\sigma_j^2+\lambda} . Recall that the SVD has eigenvalues ordered in a descending way, that is \sigma_i \geq \sigma_{i+1} .
For small eigenvalues \sigma_i it means that their contributions become less important, a fact which can be used to reduce the number of degrees of freedom. More about this when we have covered the material on a statistical interpretation of various linear regression methods.