matrix estimate. But estimator $\hat{\Sigma}$ of (2.2) is not necesarily positive definite \[for more details here see @Xue:31\]. Moreover it is hard to say whether it overcomes the over-dispersion in the sample eigenvalues. The following eigenvalue plot (Figure (2.1)) illustrates this phenomenon for a neighbourhood type (see $\S5$ for details on description of neighborhood type of covariance matrix) covariance matrix. Here we simulated random vectors from multivariate normal distribution with sample size $n=50$ and number of covariates $~p=20$.

![*Comparison of Eigenvalues of Covariance Matrices* ](sam_ei_plot.pdf){width=".8\textwidth"}

As is evident from Figure 2.1, eigenvalues of sample covariance matrix are over-dispersed as most of them are either too large or close to zero. Eigenvalues of the proposed Joint Penalty (JPEN) estimator and PDSCE (Positive Definite Sparse Covariance matrix Estimator (@Roth1:26) of the covariance matrix are well aligned with those of true covariance matrix. See $\S 5$ for detailed discussion. Another drawback of the estimator (2.2) is that the estimate can be negative definite.\
As argued earlier, to overcome the over-dispersion in eigen-spectrum of sample covariance matrix, we include eigenvalues variance penalty. To illustrate its advantage, consider $\lambda=0$. After some algebra, let $\hat{\Sigma}$ be the minimizer of (2.1), then it is given by: $$\hat{\Sigma}= 