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For a random number experiment, I want to simulate from a trivariate Normal distribution with correlation matrix.

$\left(\begin{matrix}1&a&b\\a&1&c\\b&c&1\end{matrix}\right)\quad (a,b,c > 0)$

The correlations $a$ and $b$ are parameters in my experimental setup, and the third value, $c$, I can choose somewhat freely. Obviously, for any given values of $a$ and $b$, there are values of $c$ for which the matrix is not positive definite. I am wondering if I can somehow work out the "allowed" values of $c$, if $a$ and $b$ are given.

I tried to find conditions that make all eigenvalues positive. The characteristic polynomial of the eigenvalue problem is

$x^3 - 3x^2 + x(3-a^2-b^2-c^2) + (a^2+b^2+c^2-2abc-1)$

That only told me that the problem is symmetric in $a,b,c$. But for my actual question, it didn't really get me anywhere. Any help how to approach this is much appreciated.


Answer: Thanks for the links to duplicated questions. In particular this post has the bounds on correlation I was looking for:

$ab - \sqrt{(1-a^2)(1-b^2)} \le c \le ab + \sqrt{(1-a^2)(1-b^2)}$

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