Weighted Average of Correlation Matrix

Question

Let $R$ and $Q$ be two correlation matrices of the same size and let $p\in[0,1]$. I'm trying to show that $pR+(1-p)Q$ is still a correlation matrix. I claim that $\sqrt pX+\sqrt{1-p}Y$ is a vector that gives me $pR+(1-p)Q$. Here, $X$ is the vector that gives $R$ and $Y$ is the vector that gives $Q$. Also, $X$ and $Y$ are independent. I can get the covariance part to match but I can't get the standard deviation in the denominator to match. I'm not sure why. Can someone show me how to do this?

Answer 1

All you have to show is that $S = pR +(1-p)Q$ is again positive semi-definite. Here, use the definition of positive semi-definite matrix and the linearity of the bilinear form: $$ w^TSw = w^T(pR + (1-p)Q)w = pw^TRw + (1-p)w^TQw \ge 0$$

for any real vector $w$ what concludes the proof.