TITLE: How to make a matrix positive semidefinite
QUESTION [7 upvotes]: We have a symmetric matrix $A$, with some entries specified and others not. We are trying to find the values of the unspecified entries so that the matrix $A$ becomes positive semidefinite. How can I prove that I can assume that the diagonal entries of $A$ are specified?
Intuitively, if we do not specify a  diagonal entry, say $i$th entry, we can take it to infinity. Then the positive definiteness of $A$ is equal to the positive definiteness of the new matrix $A[-i,-i]$ where we remove the $i$th column and row. I do not know how to show this mathematically.

REPLY [2 votes]: Let suppose C is non positive definite correlation matrix $$C=Q\Lambda Q^*=Q (\Lambda_+ -\Lambda_-)Q^*$$ Where  $\Lambda$ is diagonal matrix of Eigen values. Replace all negative eigen values with zero. 
Then we use the Symmetric , non negative definite matrix $\rho^2C$ with suitable value of $\rho$. Two choices of $\rho$ are $$\rho_1=tr(\Lambda)/tr(\Lambda_+) \space\space\space\space\space \rho_1=\sqrt{tr(\Lambda)/tr(\Lambda_+)}$$ User defined $\rho$ is also allowed. For +ve definite matrix $\Lambda=\Lambda_+$ and $\rho=1$ 
.Another Way is to add $C*I$ to your corr-matrix, where is C is a constant and I is an identity matrix.
Chan, Grace; Wood, Andrew T.A., An algorithm for simulating stationary Gaussian random fields, J. R. Stat. Soc., Ser. C 46, No.1, 171-181 (1997). ZBL0913.65142.