This example in Stephen Boyd's Convex Optimization book says that the following matrix is element-wise positive and therefore $x^TQx$ is semi-definite where $Q=W - \lambda_{min}(W)(I) \succcurlyeq 0$.
The example appears in Sec. 5.1.5, just before equation 5.8.
There's also no mention of what specifically is $\lambda_{min}$. If $\lambda_{min} \succcurlyeq 0$ then $Q$ should be element-wise less than 0.
For any symmetric matrix $M$, if $\lambda_1\geq \ldots\geq \lambda_n$ are the eigenvalues (which exist by the spectral theorem), then the eigenvalues of $M-\lambda_n I$ are just $\lambda_i-\lambda_n$. These are all nonnegative, so this new matrix is positive semi-definite. It has nothing to do with element-wise nonnegativity.