Are all symmetric matrices with diagonal elements 1 and other values between -1 and 1 correlation matrices?

A question for the statisticians and other math lovers: Are all symmetric matrices with diagonal elements 1 and other values between $$-1$$ and 1 correlation matrices?

• No, it must also be positive definite. – hard2fathom Jan 6 at 15:01
• @hard2fathom, thank you for your answer! What is this? – Math123 Jan 6 at 15:06
• – Julius Vainora Jan 6 at 17:47

I thought this must be asked & answered before, but cannot find it, so here it goes ... Let $$S$$ be a covariance matrix (for the algebra that follows it does not matter if it is theoretical or empirical). Let $$D$$ be a diagonal matrix with the diagonal of $$S$$. Then the correlation matrix $$R$$ is given by $$R= D^{-1/2} S D^{-1/2}$$ (how you can see this directly is explained here.)
To see that $$S$$ must be positive (semi)-definite (abbreviated posdef), let $$X$$ be a random variable with covariance amtrix $$S$$, and $$c$$ a vector. Then $$\DeclareMathOperator{\var}{\mathbb{V}ar} \var(c^T X)= c^T S c \ge 0$$ since variance is always nonnegative. Then this transfers to the correlation matrix: $$c^T R c = c^T D^{-1/2} S D^{-1/2} c = (D^{-1/2} c)^T S (D^{-1/2} c) \ge 0$$ Armed with this it is easy to make an counterexample, the following is not a correlation matrix: $$\begin{pmatrix} 1 & -0.9 & -0.9 \\ -0.9& 1 & -0.9 \\ -0.9 & -0.9 & 1 \end{pmatrix}$$