How do I specify the inverse of a correlation matrix? To specify a correlation matrix $\in \mathbf{R}^{n\times n}$. There are $n(n-1)/2$ free elements. If I wanted to specify a matrix that is the inverse of some correlation matrix, how should I specify the constraints besides it is symmetric?
My goal is to fit a correlation matrix through maximum likelihood, and the inverse appears in the objective function.  
 A: A correlation matrix must be not only symmetric but positive definite. The same goes for the inverse.
Updated: As noted in the comments, a true correlation matrix is a normalized covariance matrix, with ones in the diagonal (put in other way, it does not include the variances, only the cross-covariances). So you should take that into account. I didn't mention that because, given the context of your question (MLE) I guessed (perhaps incorrectly) that you actually meant a covariance matrix (that poor terminology is often found in signal processing texts). Perhaps you should clarify if you mean a correlation matrix or a covariance matrix.
A: *

*Every correlation matrix is a non-negative-definite (or "positive semidefinite") $n\times n$ matrix (for some $n$) in which every diagonal entry is $1$.

*Every $n\times n$ matrix that is non-negative-definite and in which every diagonal entry is $1$ is a correlation matrix.


The first bullet point above is easy to prove by using the definition of correlation and the Cauchy–Schwarz inequality.
The second is slightly harder.  Every real symmetric matrix $M$ has a spectral decomposition $M = G^\top \Lambda G$ where $G$ is an orthogonal matrix (i.e. $G^\top G = GG^\top = I$) and $\Lambda$ is a diagonal matrix with real entries.  If the matrix is non-negative-definite then $G$ can be so chosen that all diagonal entries of $\Lambda$ are non-negative.  Since they are non-negative, they have non-negative square roots.  Let $A = G^\top \Lambda^{1/2} G$ (where one just takes square roots of diagonal entries. Let $Z$ be a column vector whose entries are i.i.d. $N(0,1)$.  Then one can show that $AZ$ is a column vector whose entries are random variables with correlation matrix $M$.
To prove all the assertions in the paragraph above, one needs some linear algebra (the spectral decomposition of a symmetric matrix) and some basic results about random vectors and their covariance matrices.
