Boundaries on correlation coefficient given five other correlations Is there a general formula for the boundaries of a correlation coefficient given a set of other correlation coefficients? I have seen the formula for three random variables where two correlations are known. For example, given correlations R_ac and R_bc the boundaries of R_ab are given by:
R_ab <= R_ac*R_bc +/- sqrt[(1-R^2_ac)(1-R^2_bc)].

Is there a more general formula? Suppose I have a 4x4 correlation matrix:
---                ---
|  1                 |
|  R_ab 1            |
|  R_ac R_bc 1       |
|  R_ad R_bd R_cd 1  |
---                ---

Is there a boundary formula for R_ab if I know R_ac, R_ad, R_bc, R_bd, and R_cd?
For example, and this is a complete and total guess on my part, something like...
R_ab <= R_ac*R_cd*R*R_bd +/- [(1-R^2_ac)(1-R^2_cd)(1-R_bd)]^(1/3).

Cristian
 A: A minimal set of constraints is known as Sylvester's Criterion: all the square submatrices anchored at the upper left corner (its "leading principal minors") must have non-negative determinant.
Note that this criterion can be applied in various ways, because you are free to apply any permutation (simultaneously) to the rows and columns.  Since such a permutation yields the correlation matrix for the same variables, merely re-ordered, it is a valid correlation matrix if and only if the original matrix was valid.
As an example, consider any $n\times n$ correlation matrix $(\rho_{ij})$ with $n(n-1)/2-1$ given correlations $\rho_{ij}=\rho_{ji}$, $\rho_{ii}=1$, and one unknown correlation.  Permute the rows and columns to place that unknown value $\rho$ in the $(n,n-1)$ and $(n-1,n)$ locations, as shown here in red:
$$\pmatrix{1 & \rho_{12} & \cdots  & \cdots& \rho_{1n} \\
\rho_{21} & 1 & \cdots & \cdots & \rho_{2n} \\
\vdots & \vdots & \ddots & \vdots  & \vdots\\
\rho_{n-1,1} & \rho_{n-2,2} & \cdots & 1 & \color{red}{\rho} \\
\rho_{n1} & \rho_{n2} & \cdots & \color{red}{\rho} & 1
}$$
All the proper leading minors involve known coefficients and therefore, presumably, already have non-negative determinants.  It remains to evaluate the determinant of the entire matrix.  This is a quadratic form in $\rho$.  (This becomes obvious when you recall that the determinant is a sum and difference of products of $n$ terms of the matrix at a time, where no two terms occupy the same row or column.  Thus $\rho^2$ can appear as well as $\rho$ times other numbers, but no higher power of $\rho$ can be created.) 
Therefore the non-negativity of this determinant (along with the obvious constraint $|\rho|\le 1$) determines either an interval of possible values of $\rho$ or two intervals of the form $[-1,\rho_{-}]$, $[\rho_{+},1]$.  The endpoints are easy to compute using the Quadratic Formula.
When more than one correlation is unknown, the answer may depend on the pattern of missing correlations.  In any event, it requires solving a system of algebraic inequalities.  Thus it seems useless to attempt a general formula: each situation must be resolved on its own.  The constraints, being of degree up to $n$, can yield very messy solutions.
