Correlation matrix with all positive correlations positive definite

Question

I am trying to prove the above statement but not able to find an elegant way.... any thoughts would be appreciated.

Answer 1

That statement is not true in general.

For example,

$$\begin{pmatrix} 1 & 0.9 & 0.2 \\ 0.9 & 1 & 0.7 \\ 0.2 & 0.7 & 1 \end{pmatrix} $$

Answer 2

You cannot. The statement is false. The most obvious example is $$ \begin{pmatrix}1&1\\1&1\end{pmatrix}, $$ where the random variables are perfectly correlated. For a less obvious example, take any $n\times k$ random matrix $X$ such that (a) $n\ge k$, (b) all row sums of $X$ are zero and (c) in each column, all entries have the same sign. Then the covariance matrix $XX^T$ has deficient rank but it almost always have all positive entries. And so does the correlation matrix. For a concrete example, consider $$ X=\begin{pmatrix} 1& 0&-1& 0\\ 1& 1& 0&-2\\ 0& 2&-1&-1\\ 1& 1&-2& 0\\ 1& 1&-1&-1 \end{pmatrix}, \ XX^T=\begin{pmatrix} 2&1&1&3&2\\ 1&6&4&2&4\\ 1&4&6&4&4\\ 3&2&4&6&4\\ 2&4&4&4&4 \end{pmatrix}. $$