Conditions on correlation parameter for positive (semi) definitiveness of variance-covariance matrix

Question

Consider a 3-variate random vector normally distributed with mean $\mu$ and variance-covariance matrix $$ \Sigma=\begin{pmatrix} 1 & \rho & \rho\\ \rho & 1 & \rho\\ \rho & \rho & 1 \end{pmatrix} $$

Could you help me to understand under which conditions on $\rho$ $\Sigma$ is positive semi-definite and under which conditions on $\rho$ $\Sigma$ is positive definite?

We know that $\rho\in [-1,1]$. Since $Det(\Sigma)=1-3\rho^2+2\rho^3$, we have that $Det(\Sigma)\geq 0$ iff $\rho\geq -0.5$. Is this sufficient?

Answer 1

These two statements are equivalent

1. $\Sigma$ is positive definite

2. All eigenvalues of $\Sigma$ are positive

The eigenvalues of $\Sigma$ are $\{1-\rho, 1-\rho,1 + 2\rho\}$, which implies that we require

$$ 1 - \rho > 0 ~~~\mbox{and}~~~ 1 + 2\rho > 0 $$

or equivalently

$$ -1/2 < \rho < 1 $$

For the semi-definite case you can change "$<$" by "$\leq$"

Answer 2

This is an approach, note that you can write $\Sigma$ as $$ \Sigma = \rho e'e + (1 - \rho)I, $$ with $I$ the identity matrix and $e = (1,1,1)'$. We can take $O$, the matrix that consist of the orthonamal vectors $e/\sqrt{3},u,v$ and note that rotating perservs the question and the new sigma matrix (it preserves the eigen values) $$ \Sigma_2 = O'\Sigma O = 3 \rho \,\text{diag}(1,0,0) + (1 - \rho) I = \text{diag}(2\rho + 1, 1 - \rho, 1 - \rho) $$ And we have found all roots and the other answer apply.