Smallest possible correlation between pair of random variables

Question

Suppose $M$ is an $n \times n$ correlation matrix, with correlation $\rho$ between any pair of two random variables. What is the smallest possible value of $\rho$?

Answer 1

Let $K$ be the matrix all of whose entries are $1$; let $I$ be the identity matrix. The question is, what is the smallest $\rho$ such that $M=(1-\rho)I + \rho K$ is positive semi-definite. (All the off-diagonal entries of $M$ are equal to those of $\rho K$; addition of $(1-\rho)I$ to $\rho K$ fixes the diagonal entries, which, as orangeskid points out, are all equal to $1$.) The spectrum of $I$ is $n$ copies of $1$; the spectrum of $K$ is $n-1$ copies of $0$ and one of $n \rho$; that of $M$ is $n-1$ copies of $1-\rho$ and one copy of $1-\rho+n\rho=1-(n-1)\rho$. So the condition on $\rho$ is that $1-\rho\ge0$ and $1-(n-1)\rho\ge0$, that is, $1\ge\rho\ge-1/(n-1)$.