TITLE: How to find closest positive definite matrix of non-symmetric matrix
QUESTION [10 upvotes]: I have a matrix $A$ given and I want to find the matrix $B$ which is closest to $A$ in the frobenius norm and is positiv definite. $B$ does not need to be symmetric.
I found a lot of solutions if the input matrix $A$ is symmetric. Are they any for a non-symmetric matrix $A$? Is it possible to rewrite the problem as a minimization of a convex problem?

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.