While researching a question, I faced with the following problem: I have to prove that for a positive definite matrix we have

$${\mathbf n}^T {\mathbf R}^{-1}{\mathbf n}\geq {\mathbf n}^T {\mathbf D} {\mathbf n}$$ for all ${\mathbf n}$, where ${\mathbf R}$ is a $K$ times $K$ positive definite matrix and the diagonal matrix $\mathbf{D}$ is defined as

$${\mathbf D} \triangleq \frac{1}{K} ({\mathbf R} \odot \mathbf{I})^{-1}$$

and $\odot$ is elementwise product (Hadamard product), ${\mathbf I}$ is the identity matrix. It looks like a classic problem which provides a bound for the inverse matrix and the inverse of its diagonal elements. Is there any proof for this bound?

So far I have realized that may be the bounds for matrix norm would be useful.